# Magnetic and structural properties of the iron silicide superconductor
LaFeSiH

M. F. Hansen  CNRS, Université Grenoble Alpes, Institut Néel, 38042 Grenoble,
France  S. Layek  CNRS, Université Grenoble Alpes, Institut Néel, 38042
Grenoble, France  Department of Physics, School of Advanced Engineering,
University of Petroleum and Energy Studies, Dehradun, Uttarakhand 248007,
India  J.-B. Vaney  Univ. Bordeaux, CNRS, Bordeaux INP, ICMCB, UMR 5026,
F-33600 Pessac, France  L. Chaix  CNRS, Université Grenoble Alpes, Institut
Néel, 38042 Grenoble, France  M. R. Suchomel  Univ. Bordeaux, CNRS, Bordeaux
INP, ICMCB, UMR 5026, F-33600 Pessac, France  M. Mikolasek  G. Garbarino  A.
Chumakov  R. Rüffer  ESRF-The European Synchrotron, CS40220 38043 Grenoble
Cedex 9 France  V. Nassif  CNRS, Université Grenoble Alpes, Institut Néel,
38042 Grenoble, France  Institut Laue -Langevin, 71 Avenue des Martyrs, 38000
Grenoble, cedex 9 France  T. Hansen  Institut Laue -Langevin, 71 Avenue des
Martyrs, 38000 Grenoble, cedex 9 France  E. Elkaim  Synchrotron SOLEIL, L’Orme
des Merisiers Saint-Aubin, France  T. Pelletier  CNRS, LNCMI, Univ. Grenoble
Alpes, INSA-T, UPS, EMFL, Grenoble, France  H. Mayaffre  CNRS, LNCMI, Univ.
Grenoble Alpes, INSA-T, UPS, EMFL, Grenoble, France  F. Bernardini
Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria,
I-09042 Monserrato (CA), Italy  A. Sulpice  M. Núñez-Regueiro  P. Rodière  A.
Cano  CNRS, Université Grenoble Alpes, Institut Néel, 38042 Grenoble, France
S. Tencé  Univ. Bordeaux, CNRS, Bordeaux INP, ICMCB, UMR 5026, F-33600 Pessac,
France  P. Toulemonde  CNRS, Université Grenoble Alpes, Institut Néel, 38042
Grenoble, France  M.-H. Julien  CNRS, LNCMI, Univ. Grenoble Alpes, INSA-T,
UPS, EMFL, Grenoble, France  M. d’Astuto  CNRS, Université Grenoble Alpes,
Institut Néel, 38042 Grenoble, France

(August 28, 2024)

###### Abstract

The magnetic and structural properties of the recently discovered
pnictogen/chalcogen-free superconductor LaFeSiH ($T_{c}\simeq 10$ K) have been
investigated by 57Fe synchrotron Mössbauer source (SMS) spectroscopy, x-ray
and neutron powder diffraction and 29Si nuclear magnetic resonance
spectroscopy (NMR). No sign of long range magnetic order or local moments has
been detected in any of the measurements and LaFeSiH remains tetragonal down
to 2 K. The activated temperature dependence of both the NMR Knight shift and
the relaxation rate $1/T_{1}$ is analogous to that observed in strongly
overdoped Fe-based superconductors. These results, together with the
temperature-independent NMR linewidth, show that LaFeSiH is an homogeneous
metal, far from any magnetic or nematic instability, and with similar Fermi
surface properties as strongly overdoped iron pnictides. This raises the
prospect of enhancing the $T_{c}$ of LaFeSiH by reducing its carrier
concentration through appropriate chemical substitutions. Additional SMS
spectroscopy measurements under hydrostatic pressure up to 18.8 GPa found no
measurable hyperfine field.

Magnetism,Superconductivity, Mössbauer Spectroscopy, Local Magnetic
Measurements, Synchrotron radiation

##  I Introduction

Following the discovery of Fe-based superconductors in 2008, the results of
intense research effort has unearthed many compounds, leading to what now
forms a complex and rich family of unconventional superconducting materials
Hosono and Kuroki (2015); Martinelli  _et al._ (2016); Iimura and Hosono
(2020). In these systems, the superconducting instability is believed to be
intimately linked to antiferromagnetic (AFM) fluctuations.

Here, we focus on the compound LaFeSiH, for which superconductivity has been
revealed, with $T_{c}=10$ K Bernardini  _et al._ (2018), i.e. at a value
rather similar to other superconducting Fe-silicides discovered more recently,
LaFeSiFxVaney  _et al._ (2022) and LaFeSiO1-δ Hansen  _et al._ (2022). LaFeSiH
was reported to undergo an orthorhombic distortion at moderate pressures and
low temperatures Bernardini  _et al._ (2018). According to first-principles
calculations, superconductivity in LaFeSiH is unconventional in the sense that
$T_{c}$ cannot be explained by the conventional electron-phonon pairing
mechanism Hung and Yildirim (2018). Furthermore, the measured magnetic
penetration depth reveals the emergence of a $d$-wave superconducting gap in
this system Bhattacharyya  _et al._ (2020). This circumstance can be related
to the increased orbital overlap along the $c$-axis together with the change
in the orbital content of the Fermi surface compared to the reference LaFeAsO,
which otherwise should favor the $s_{\pm}$-wave state. At the same time,
despite the reduced Fermi-surface nesting, first-principles calculations
suggest that the tendency towards magnetism in LaFeSiH is quite similar to
LaFeAsO Bernardini  _et al._ (2018); Hung and Yildirim (2018); P. Villar
Arribi  _et al._ (2020) as well as the loss of superconductivity at moderate
pressures, associated with the decrease of the magnetic moment on iron, with
an additional re-entrant behaviour as pressure is increased.

In this paper we investigate the magnetic properties of LaFeSiH as a function
of temperature, magnetic field, and pressure using SMS (57Fe Synchrotron
Mössbauer Source), NPD (neutron powder diffraction), and NMR (nuclear magnetic
resonance spectroscopy), as well as the low temperature structural properties
using XRD (X-ray diffraction).

Our measurements show no long range magnetic ordering or local moments within
detection limits Indeed, comparison of our present results using SMS in
LaFeSiH with 57Fe Mössbauer literature for Fe-based superconductorsKlauss  _et
al._ (2008); McGuire  _et al._ (2008); Kitao  _et al._ (2008); Mizuguchi  _et
al._ (2010) suggests a upper bound of 0.04 $\mu_{B}$ for an ordered magnetic
moment, in agreement with NMR which, under similar hypothesis, suggest an
upper bound of $\sim$ 0.01 $\mu_{B}$. This is well below the value of (DFT)
first-principles calculations (1.2 $\mu_{B}$) Bernardini  _et al._ (2018).
Furthermore, SMS spectroscopy do not find any re-entrant magnetism at higher
pressure up to 18.8 GPa. Based on the temperature dependence of 1/T1, and
consistent with the absence of magnetism, LaFeSiH seems located in the
overdoped regime.

##  II Methods

###  II.1 Samples

LaFeSi precursor powders were first prepared by arc melting of a
stoichiometric mixture of pure elements (La, Fe and Si), subsequently grinded,
compacted, and subjected to a thermal treatment of 7 days at 950 ∘C. The final
LaFeSiH powders were obtained from the hydrogenation of the precursors,
treated at 250 ∘C for 4 h under a static pressure of gaseous H2 of 10 bars.
The deuterated sample, LaFeSiD, for the NPD experiment was obtained with a
similar protocol, hydrogen gas being replaced by deuterium gas in the last
stage (see also Hansen  _et al._ (2023)). For the preparation of LaFeSiH
single crystals, small single crystals of LaFeSi were separated from the bulk
of an arc-melted Si-under-stoichiometric LaFeSi0.86 composition. The 20 $\mu$m
thick plate-like single crystals of various sizes were then hydrogenated at
250 ∘C under a flow of H2 gas for 4 h. Both the LaFeSiH powders and single
crystals were checked by X-ray diffraction and refinement of the unit cell
parameters were found to be in good agreement with previously reported data
Bernardini  _et al._ (2018). For the SMS measurements, additional samples were
synthesized using 96% isotopic Fe57 instead of the naturally abundant isotope.
Bulk superconductivity was confirmed by measuring the magnetization of all
samples, as described in Appendix C. Furthermore, the superconducting state
was verified using resistivity measurements for the isotope enriched sample,
as described in Appendix B.

###  II.2 SMS spectroscopy under pressure and magnetic field

Standard 57Fe Mössbauer spectroscopy with laboratory sources has been
extensively applied to investigate structural, electronic and magnetic
properties of iron-based pnictides and chalcogenides Kamusella  _et al._
(2017). In order to probe the magnetic phase diagram of LaFeSiH at both
ambient and high pressure, we instead choose SMS spectroscopy using a
synchrotron source to allow the combination of both low temperature, high
pressure and high magnetic field. While combining high-pressure with low
temperature and high magnetic field bears no novelty using standard Mössbauer
spectroscopy with laboratory sources Fultz (2011) or Nuclear Forward
Scattering Lübbers  _et al._ (1999), the possibility to use SMS spectroscopy
at these conditions has emerged only recently, with relatively few
publications relating this option Adler  _et al._ (2019, 2020, 2022). This
despite the advantage of having a very focused beam, already in use since its
beginning for standard high pressure experiments Potapkin  _et al._ (2012). We
used the 57Fe Synchrotron Mössbauer Source (SMS) Smirnov  _et al._ (1997);
Potapkin  _et al._ (2012) at the Nuclear Resonance beamline (ID18) Rüffer and
Chumakov (1996) of the European Synchrotron Radiation Facility (ESRF,
Grenoble), where this was developed. The 57FeBO3 crystal was set in the (111)
reflection and sinusoidal acceleration configuration as described in Ref.
Potapkin  _et al._ , 2012. Energy calibrations were performed for each
velocity configuration of the Doppler spectrometer, using a 25 $\mu$m thick
natural $\alpha$-iron absorber standard. We used a maximum velocity of
11.30(2) mm/s for the first set of measurements, at low field and ambient
pressure, and 5.64(1) mm/s for the remaining measurements. Here we detail the
measurements of two samples, one with isotope enriched iron and one with
natural iron. This is further detailed in Appendix A.

Data analysis was carried out with the MossA procedure for Matlab©Prescher
_et al._ (2012).

An example of the data obtained in LaFeSiH at low temperature is shown in Fig.
1, at ambient pressure and without any applied field, as further detailed in
Sec. III.1 (raw data are shown in Fig. 7, see Appendix A).

For all measurements the cryomagnet system of the beamline was used, providing
temperatures between 2 and 300 K and an external magnetic field between -8 and
+8 T in vertical direction. The system is equipped with aluminum windows for
the $\gamma$-ray beam and quartz windows for the ruby fluorescence calibration
signal. Pressure calibration was performed in-situ after each change of
pressure and/or temperature conditions.

A membrane driven diamond anvil cell (DAC) with 500 $\mathrm{\mu}$m culet
diamonds and stainless steel gasket was used. The enriched and natural sample
were loaded into the same DAC, used for the temperature and field measurements
at ambient pressure (as it can be seen in Fig. 8 in Appendix A), using He as a
pressure-transmitting medium. Using an optics system with a very long working
distance (f $\frac{1}{28}$ 50 cm) we measured in-situ the ruby fluorescence
for pressure calibration.

###  II.3 Diffraction techniques

Neutron powder diffraction (NPD) was carried out at the D2B powder diffraction
instrument of the Institute Laue Langevin (ILL) using a wavelength of
$\lambda$ = 1.5944 Å. The deuterated sample, LaFeSiD, was inserted in a
cylindrical vanadium sample holder and measured at 10 fixed temperature points
in the 10 K - 300 K range with an exposure time of $\sim$ 2h per point and
$\sim$ 4 h at 2 K. The empty cryostat was also measured, showing only one
significant background peak at 151.2 deg. X-Ray powder diffraction (XRD) was
carried out at the CRISTAL beamline of the synchrotron SOLEIL. The sample was
loaded in a 300 $\mu$m diameter borosilicate capillary. Two different
detectors were used in parallel for each: a fast photon counting strip
detector (Mythen2) for medium angular resolution but excellent counting
statistics acquisition and a multi-crystal analyser for high angular
resolution and medium counting statistics acquisition. Precise lattice
parameters were determined using the multi-crystal analyser data and the
accurate atomic positions were obtained from refinements of the Mythen2 data.
The wavelength used for the experiment was $\lambda$ = 0.51302 Å. The
collected data was corrected for absorption using a $\mu$R value of 1.23. XRD
and NDP results are compared in appendix D.

###  II.4 Nuclear magnetic resonance

29Si NMR experiments were performed in an external field of $\sim$15 T (the
exact field value is calibrated from the 63Cu NMR resonance frequency in
metallic Cu) down to 2.85 K. At this field, the superconducting transition is
not detected in our measurements so only the normal state is probed here. We
have measured three NMR observables:

1) The magnetic hyperfine shift $K$, also called Knight shift, which is
calculated from the frequency difference between the measured 29Si resonance
and the calculated resonance of the bare nucleus. $K$ is usually written as
the sum of a $T$ independent orbital part and a temperature dependent spin
part : $K=K_{\rm orb}+K_{\rm spin}$, where $K_{\rm spin}$ is proportional to
the local, uniform $(q=0)$ and static ($\omega=0)$ spin susceptibility
$\chi_{\rm spin}$.

2) The line width, providing a measure of the spatial distribution of
hyperfine shifts that may result from chemical, lattice and electronic
inhomogeneity on all length scales. In a random powder, the Knight shift
anisotropy also contributes to the line width but since the line shape does
not show any shoulder or splitting here (Fig. 6a), the line broadening turns
out to be rather dominated by spatial inhomogeneity. The line shape has been
fit by a Voigt profile, that is, a convolution of Gaussian and Lorentzian
functions of widths $w_{G}$ and $w_{L}$, respectively, giving a full width at
half maximum FWHM = $0.5346\,w_{L}+\sqrt{0.2166\,w^{2}_{L}+w^{2}_{G}}$.

3) The spin-lattice relaxation rate, $T_{1}^{-1}$, providing information on
low-energy spin dynamics through the imaginary part of the dynamic spin
susceptibility:
$(T_{1}T)^{-1}\propto\sum_{q}A_{\perp}^{2}(q)\chi^{\prime\prime}_{\perp}(q,\omega_{n})$,
where $\omega_{n}$ is the nuclear Larmor frequency ($\sim 10^{8}$ Hz here) and
$A(q)$ a site-specific hyperfine form factor. $T_{1}$ values were determined
from single-exponential fits of the time dependence of the nuclear
magnetization following a saturating pulse. Excellent fits were obtained at
all temperatures, without introducing any stretching exponent.

##  III Experimental results and discussion

###  III.1 SMS spectroscopy

In Fig. 1, we show SMS spectra of 57Fe isotope in LaFeSiH taken at low
temperature, at 12 K and 3 K, at ambient pressure and without any applied
magnetic field. The spectra fitted well by a single line, with possible
quadrupolar splitting, at best of the same order of the resolution, of about
0.1 mm/s, also indicating a very weak electric field gradient on the iron
site.

Figure 1:  (Color online) Synchrotron 57Fe Mössbauer of LaFeSiH natural iron
sample at 3 K (superconducting state) and 12 K (normal state) at ambient
pressure. Empty circles represent experimental data points. The Mössbauer
spectra are fitted with single line (cyan fill) plus an additional
contribution, also modeled with a single line (orange fill) coming from iron
impurities in Beryllium window and collimating lenses of the beamline. The
blue solid line through the data points represents the overall fit convoluted
with the instrumental resolution.

Thus we detect no magnetic hyperfine field on the iron site within the
experimental resolution, which for an ordering of Fe magnetic moment, would
have been expected to give magnetic splitting. However, a very weak magnetic
moment would induce a small splitting, so, a single line shape can not
guarantee a zero moment. The instrumental line width and distributions such as
isomer shift and quadrupole splitting may also broaden the single line. We
estimate an upper bound of the magnetic hyperfine field being $\sim$ 0.6 T.

As a comparison, in LaFeAsO, 57Fe Mössbauer spectroscopy measures a hyperfine
magnetic field Bhf of 4.86 T, according to Ref. Klauss  _et al._ , 2008,
giving an estimate of the magnetic moment of m(Fe) = 0.25(5) $\mu_{B}$ at 13
K, while Ref. Kitao  _et al._ , 2008 and McGuire  _et al._ , 2008 found about
5.3-5.2 T giving a larger estimate of the magnetic moment of about 0.35
$\mu_{B}$ at a lower temperature of 4.2 K. Despite a difference in the
estimation, both values are far above the value we find in LaFeSiH as can be
easily seen in the spectra, which clearly shows a magnetic spliting in LaFeAsO
at low temperature, contrary to the LaFeSiH case.

Without a precise knowledge of the electronic and magnetic state of the
compound, it is difficult to estimate a upper limit for the magnetic moment in
our case, however we can estimate an upper limit of an ordered magnetic moment
on iron, in the hypothesis of a similar magnetic structure, below $\sim$ 0.04
$\mu_{B}$. Indeed, the observed signal is closer to what is measured in
tetragonal FeSe Mizuguchi  _et al._ (2010) as well as doped, superconducting
LaFeAsO0.89F0.11 Kitao  _et al._ (2008).

Figure 2:  (Color online) (a) Empty black circles represent SMS spectra of
LaFeSiH natural iron sample at ambient pressure and 12 K at different applied
magnetic field from 0 to 7 T whereas the blue solid line through the data
points represents the overall fit convoluted with the instrumental resolution.
Mössbauer spectra were fitted with a magnetic splitting (green) plus a single
line, orange filled, coming from iron impurities in Beryllium window and
collimating lenses of the beamline. (b) The extracted hyperfine field
$B_{hf}(H)$ (green squares), compared to a linear fit (red dot-dashed line).

In order to investigate a possible localized, disordered moment, we applied a
magnetic field, H, up to 7 T. In Fig. 2, top panel, we show the results
ramping up from zero to 7 T for the natural iron sample at 12 K and ambient
pressure. Similar results where obtained for the field downstroke on the
enriched iron sample. At the maximum, 7 T, field an additional measurement on
a second enriched sample gave the same Bhf within error bars. In Fig. 2,
bottom panel, we show the fitted Bhf as a function of the applied field $H$,
fitted by linear regression. The small deviation from the applied field is
constant on all the data points for the field upstroke. Our fit gives a slope
of 0.98 $\pm$ 0.03, consistent with unity within the error bars, pointing to
an effect coming exclusively from the applied field without any contribution
from a molecular field acting on the 57Fe nuclei. A similar observation was
made for LaFeAsO0.89F0.11 Kitao  _et al._ (2008), which also shows an
hyperfine field at 4.2 K corresponding within error-bars to the applied
external field at 7 T.

Figure 3:  Synchrotron 57Fe Mössbauer spectra of LaFeSiH natural iron sample
as a function of pressure up to 18.8 GPa, in zero field and at the lowest
temperature for each pressure point. Exact values of pressure and temperature
are written in the respective plots. For each pressure, the temperature of the
shown data is always above the superconducting one. Note that the data at
ambient pressure are taken outside the DAC cell. Same symbols and color code
as in Fig. 1.

We note that measurements performed during applied field downstroke on
enriched iron sample with 96% 57Fe, give results that are perfectly superposed
to the upstroke between 5 and 7 T, i.e. as long as a magnetic splitting is
larger than the intrinsic line-width, which is larger for enriched iron sample
because of its broadened signal, compared with natural iron sample.

We then applied a similar procedure on the same sample, loaded in the high
pressure diamond anvil cells as described in Sec. II.2. We investigated the
Mössbauer signal up to 18.8 GPa, and down to 6 K, where we have only performed
a zero field measurements, before breaking the diamonds while attempting to
reach higher pressure. A summary of the results is shown in Fig. 3. In
LaFeAsO, pressure monotonically decreases both TN and $B_{hf}$, while
initially increasing Tc from zero up to $\sim$ 20 K at about 10 GPa Kawakami
_et al._ (2009). The results shown here in LaFeSiH are similar to the ones at
ambient pressure, although a broadening is now observed, that can be fitted
with a residual hyperfine field at zero applied field. However, we found that
all measurements in the cell give a large broadening of the lineshape,
possibly due to vibration of the high-pressure membrane DAC as well as a
distribution of the pressure inside the sample. In particular, for the highest
pressure at zero field, we estimate a maximum possible hyperfine field of 1.3
$\pm$ 0.3 T, larger than the value found in samples outside the DAC cell, but
with a line-shape that can be perfectly interpreted as coming from a single
line.

In summary, the Mössbauer data, does not show any splitting of the nuclear
resonance which can be linked to a hyperfine magnetic field from a magnetic
polarization of the electronic cloud on iron sites, neither at ambient
pressure and magnetic field, nor at high pressure, up to $\approx$ 19 GPa.

Measurements under field give a hyperfine field that corresponds to the
applied field within error bars, up to 7 T, both at ambient pressure (Fig. 2)
and up to $\approx$ 12 GPa (see Fig. 9 in Appendix A). This rules out a
disordered magnetic state with a large, $\approx$ 1 $\mu_{B}$ moment.

The present results can not, however, rule out a small moment and/or a very
complex (and unusual) order which minimizes the hyperfine field by symmetry.
We note that a very small moment, below m(Fe) $\approx$ 0.05 $\mu_{B}$, in a
disordered state, could be possibly probed using X-ray emission spectroscopy,
looking at the Kβ line satellite, a very sensitive probe to partially filled
$3d$ electronic band, as recently shown in $\epsilon$-iron Lebert  _et al._
(2019) and FeSe Lebert  _et al._ (2018). Another possibility would be to use
synchrotron radiation perturbed angular correlation spectroscopy, another
approach recently applied to $\epsilon$-iron Bessas  _et al._ (2020).

###  III.2 Neutron powder diffraction

Fig. 4 shows NPD patterns recorded at 300 K and 2 K. No splitting or
additional Bragg peaks are observed at low temperatures, the peaks are only
shifted in angle due to the thermal contraction of the unit cell. This is
consistent with the absence of a symmetry breaking transition and absence of
any long range magnetic order with k$\neq$0 at low T. No systematic extra
intensity is observed on top of the nuclear diffraction peaks, except changes
of relative intensities related to modifications of the atomic positions in
the cell. This suggests a lack of long range magnetic ordering related to k=0.
Thus, our NPD data shows no sign of long range magnetic ordering with a
significant magnetic moment m(Fe).

Figure 4: Neutron powder diffraction patterns at 300 K (in red) and 2 K (in
blue) recorded at the D2B instrument at ILL. No additional peaks are observed
between the two temperatures, consistent with the lack of long range magnetic
ordering with k$\neq$ 0\. Peaks indicated by a star originate form a small
La(OH)3 impurity. Inset: Neutron powder diffraction pattern measured at 2 K.
Simulations of the single stripe AFM structure is plotted on top of the data,
with different magnetic moment values ranging from m(Fe) = 1.2 $\mu_{B}$
(blue) to m(Fe) = 0.1 $\mu_{B}$ (green).

To estimate the detection limit of the NPD experiment we performed several
simulations. We used the usual single-stripe-type AFM structure found in
LnFeAsO arsenidesde la Cruz  _et al._ (2008), with varying moments from m(Fe)
= 1.2 $\mu_{B}$ (i.e. the value obtained from DFT P. Villar Arribi  _et al._
(2020)) to m(Fe) = 0.1 $\mu_{B}$, taking into account an orthorhombicity of
$\delta{}=(b-a)/(b+a)\sim 2\leavevmode\nobreak\ 10^{-3}$ i.e. the magnitude
measured in DAC at moderate pressures Bernardini  _et al._ (2018). The
calculated pattern of the nuclear model, issued from the Rietveld fit of the
observed pattern collected at 2 K, is shown in red in the inset of figure 4.
By comparing our simulations to the noise level of the data we can estimate
that a magnetic order with moments below m(Fe) = $0.15(5)\mu_{B}$ would not be
detected in our experiment. This can thus be considered as an upper bound for
m(Fe) from NPD.

###  III.3 X-ray powder diffraction

Figure 5: The refined structural parameters as a function of temperature,
obtained from synchrotron x-ray diffraction (at the CRISTAL beamline at
SOLEIL). a: The unit cell parameter a and c. b: The unit cell parameter ratio
c/a. c: The fractional z-coordinates of Si and La. d: The FeSi4 tetrahedra
angles $\alpha$ and $\beta$. e: The Fe-Si height and bond length. f: In red,
the La-Si height. In black, c/2. This displays the contraction mechanism upon
cooling.

In comparison to the NPD experiment, the higher angular resolution of our XRD
experiment allows us to confirm that LaFeSiH does not undergo any symmetry
lowering structural distortion, as also shown by Raman spectroscopy Layek  _et
al._ (2023). This is in contrast to the previously reported tetragonal to
orthorhombic distortion detected at 15 K and at moderate pressures Bernardini
_et al._ (2018). This is however, consistent with the absence of long range
magnetic order, since such structural distortion is usually observed before
the onset of stripe-type anti-ferromagnetic order in Fe-based superconductors
Qureshi  _et al._ (2010). In figure 5.a we show the refined lattice
parameters, a and c respectively, obtained from Rietveld fits of the XRD data
using a tetragonal model (P4/nmm). Firstly, the thermal contraction of the
unit cell with temperature is clearly evident for the c parameter. The a
parameter, however, displays an almost static behavior except for a minute
broad minimum around 175 K. This minimum, possibly related to the coupling of
the lattice with an electronic degree of freedom, was also observed in the
refined lattice parameters from NPD (See appendix D). The origin of such
behavior can not be readily explained by this data. To uncover the underlying
mechanism of this anomaly, further studies are required which are out of the
scope of this work. The refinable atomic positions in the unit cell, z(Si) and
z(La), are shown in figure 5.c. Both evidence an asymptotic increase with
decreasing temperature. This results in the Fe-Si height and Fe-Si bond length
remaining nearly unchanged in the entirety of the temperature range, as shown
in figure 5.e. The relative local environment of the FeSi4 tetrahedron also
remains unmodified, as shown by the temperature dependence of alpha and beta
angles in 5.d. The contraction of the c-axis can be attributed solely to the
decrease of the La-Si height, i.e. the inter-block distance. In figure 5.f
this distance is plotted along with c/2 to illustrate the contraction
mechanism.

###  III.4 Nuclear magnetic resonance

Figure 6:  (a) 29Si NMR line (green) in a field of $\sim$15 T and $T=2.85$ K.
The thin black line is a fit to a Voigt profile. (b) Full width at half
maximum (FWHM). The horizontal line is a fit to a constant value 45$\pm$3 kHz.
(c) Total magnetic hyperfine shift $K$. The line is a fit to an activated
dependence with a gap value $\Delta_{K}=106\pm 24$ K. The negative sign of $K$
with larger absolute values at high temperatures shows that the hyperfine
coupling of 29Si is negative. (d) Spin-lattice relaxation rate $T_{1}^{-1}$
divided by $T$. The continuous and dashed lines are fits to an activated
dependence with gap values $\Delta_{T_{1}}=101\pm 38$ K (reduced
$\chi^{2}=0.61$) and $\Delta_{T_{1}}=2\Delta_{K}=212$ K (fixed, reduced
$\chi^{2}=1.33$), respectively. Although the data does favor
$\Delta_{T_{1}}\simeq\Delta_{K}$, $\Delta_{T_{1}}\simeq 2\Delta_{K}$, as
observed in LaNiAsO1-xFx Tabuchi  _et al._ (2010), cannot be fully excluded.
(e) Korringa ratio $\alpha_{\rm
Korr}=\hbar\,\gamma_{e}^{2}/(4\,\pi\,k_{B}\,\gamma_{n}^{2})\,(K^{2}T_{1}T)^{-1}$.
The thick trace is a guide to the eye. (f) The nearly $T$ independent
$(KT_{1}T)^{-1}$.

In general, magnetic order has two main effects on NMR observables. 1) It
either shifts, splits or broadens the NMR lines (except in those exceptional
cases where the hyperfine field vanishes at the nucleus position for symmetry
reasons, which should not be the case for 29Si here). 2) It produces a sharp
change in the spin-lattice relaxation rate $T_{1}^{-1}$, either an abrupt drop
upon entering the ordered state if the transition is first order or a peak at
the transition temperature if the transition is second order (and also a peak
for a continuous freezing).

None of these features are observed in our 29Si NMR data in LaFeSiH: all
observables measured in our 29Si NMR data of LaFeSiH show weak and smooth
temperature dependence (Fig. 6), which rules out the presence of magnetic
order.

Specifically, the width of the 29Si NMR line stays constant from room
temperature down to 2.8 K (Fig. 6b). Given that the conjunction of electron
correlations and (unavoidable) disorder tends to broaden the NMR lines upon
cooling, the constant linewidth here indicates that spin correlations are
actually very weak, if any. The absence of stretched-exponential behavior of
the magnetization recovery shows that the electronic spin polarization remains
spatially homogeneous at all temperatures, which is also consistent with the
weakness of spin correlations (even moderate antiferromagnetic correlations in
the sister compound LaFeSiO induce stretched-exponential behavior Hansen  _et
al._ (2022)). The constant linewidth here is also consistent with the absence
of an orthorhombic transition inferred from XRD (orthorombicity differentiates
$K$ values along $a$ and $b$ axes, which splits the NMR lines, see Ref. Zhou
_et al._ (2020) and references therein). However, the relatively large width
in the powder sample obviously makes NMR a much less sensitive probe than XRD
in the present case and NMR cannot exclude small and/or short-range
distortions due to pinned nematic fluctuations Wiecki  _et al._ (2021).

Both the Knight shift (Fig. 6c) and $(T_{1}T)^{-1}$ (Fig. 6d) decrease
smoothly upon cooling and saturate to a constant value below $\sim$20 K. A fit
to an exponential activation $a+b\times\exp(-\Delta/k_{B}T)$ yields virtually
identical gap values in the Knight shift ($\Delta_{K}=106\pm 24$ K) and in
$T_{1}$ ($\Delta_{T_{1}}=101\pm 38$ K).

At the qualitative level, the exponential dependence of both $K$ and
$(T_{1}T)^{-1}$ over the whole $T$ range is typical of strongly overdoped Fe-
based superconductors Yang  _et al._ (2015); Tabuchi  _et al._ (2010); Ma  _et
al._ (2011). At the quantitative level, our data strikingly resembles results
in strongly overdoped Ba(Fe1-xCox)2As2 for which
$\Delta_{T_{1}}\simeq\Delta_{K}$ is also observed Ning  _et al._ (2010). The
thermally-activated $(T_{1}T)^{-1}$ has been ascribed to spin fluctuations
associated with small momentum transfers (so-called ”intra-band” scattering)
Ning  _et al._ (2010). Note that this contribution is also present in the
underdoped regime but it becomes increasingly masked upon cooling as the
growth of $(\pi,0)$ spin fluctuations (”inter-band” scattering) leads to a
large upturn in $(T_{1}T)^{-1}$. Ma and Yu (2013); Carretta and Prando (2020)
Such stripe-type spin fluctuations are manifestly absent in LaFeSiH.

Assuming that the hyperfine coupling of 29Si in LaFeSiH is similar to that of
31P in BaFe2(As1-xPx)2 (0.61 T/$\mu_{B}$, ref. Iye  _et al._ (2012)) and
considering that our detection limit is set by the linewidth of 45 kHz, an
approximate upper bound for an ordered moment is $\sim 0.01\mu_{B}$. This is
consistent with the observation that an ordered moment of 0.05 $\mu_{B}$ in
BaFe2(As1-xPx)2 ($x\simeq 0.3$) Hu  _et al._ (2015) leads to $T_{1}$ results
for 31P that are very different Hu  _et al._ (2015); Iye  _et al._ (2012) from
those observed here.

The Korringa ratio
$\hbar\,\gamma_{e}^{2}/(4\,\pi\,k_{B}\,\gamma_{n}^{2})\,(K^{2}T_{1}T)^{-1}=\alpha_{\rm
Korr}$ (where $\gamma_{e}$ and $\gamma_{n}$ are the electron and nuclear
gyromagnetic ratio, respectively) is found to be slightly enhanced with
respect to $\alpha_{\rm Korr}=1$ (Fig. 6e). This points to rather weak
electron correlations in LaFeSiH, consistent with the absence of both magnetic
order and stripe-type magnetic fluctuations (notice that we implicitly assume
here that $K$ is relatively isotropic and that $K_{\rm orb}\ll K_{\rm spin}$
so that $K\simeq K_{\rm spin}$). Nonetheless, $\alpha_{\rm Korr}$ is not
constant as a function of $T$, indicating that the Korringa law of Fermi
liquids $K_{\rm spin}^{2}\propto(T_{1}T)^{-1}$ is not satisfied. Instead,
$(K\,T_{1}T)^{-1}$ is nearly $T$ independent (Fig. 6f).

We thus conclude from this NMR data that LaFeSiH probably has a Fermi surface
topology close to that of strongly overdoped iron-pnictides, with neither spin
order nor even stripe-type fluctuations. This raises the prospect of enhancing
the $T_{c}$ of LaFeSiH by appropriate chemical substitutions. These results
contribute to establishing a solid, quantitative phenomenology of NMR data in
overdoped Fe-based superconductors that could be confronted to theoretical
predictions based on realistic band structures. Another interesting prospect
on the theory side is whether ab-initio calculations are able to account for
the negative sign of the hyperfine coupling of 29Si in LaFeSiH (as shown by
the negative values of $K$ in Fig. 6c). This is opposite to the hyperfine
coupling of 29Si in LaFeSiO, a sister compound with strongly squeezed FeSi
layers Hansen  _et al._ (2022).

##  IV Conclusions

In summary, from our SMS, NMR and NPD experiments we do not observe any long
range magnetic order or local moments within detection limit in LaFeSiH at
ambient pressure, down to 2 K. Comparison of our present results using SMS in
LaFeSiH with 57Fe Mössbauer literature for Fe-based superconductorsKlauss  _et
al._ (2008); McGuire  _et al._ (2008); Kitao  _et al._ (2008); Mizuguchi  _et
al._ (2010) suggests a upper bound of 0.04 $\mu_{B}$ for an ordered magnetic
moment, in agreement with NMR which, under similar hypothesis, suggest an
upper bound of $\sim$ 0.01 $\mu_{B}$. Furthermore, our high-pressure SMS
measurements do not indicate any magnetic moment up to 19 GPa and down to 6 K.
From our XRD and NPD experiments we observe no indication of a lattice
distortion into an orthorhombic setting down to 2 K, while the resolution of
our experiment would allow us to easily detect distortions of the magnitude
previously claimed in LaFeSiH Bernardini  _et al._ (2018) or the one observed
in the related compound LaFeAsO de la Cruz  _et al._ (2008).

###### Acknowledgements.

We thank M.-A. Méasson, I. Vinograd and R. Zhou for discussions. Part of this
work was performed at the LNCMI, a member of the European Magnetic Field
Laboratory (EMFL). This work was supported by the ANR-18-CE30-0018-03 Ironman
grant. We acknowledge the European Synchrotron Radiation Facility for
provision of synchrotron radiation resources at the Nuclear Resonance beamline
(ID18). We acknowledge the synchrotron SOLEIL for providing synchrotron
radiation resources at the beamline CRISTAL. We acknowledge the Institute Laue
Langevin (ILL) for providing neutron radiation at the D2B instrument.

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##  Appendix A SMS specroscopy additional information

Figure 7:  (Color online) Synchrotron 57Fe Mössbauer raw data of LaFeSiH
natural iron sample at 11.80(5) K (normal state) and 2.96(5) K
(superconducting state) at ambient pressure as in Fig. 1. The spectra
accumulation times were 1061 sec at 3K and 852 sec at 12 K. Same symbols and
color code as in Fig. 1.  Figure 8:  (Color online) Single crystals S #4 with
96% 57Fe substitution and S #7 with natural iron loaded in the high-pressure
cell, together with the ruby pressure gauge.

We measured 7 crystals at ambient pressure, 2 with natural iron and the
remaining 5 highly enriched in 57Fe. Based on these measurements we selected
two of them, one with 96% 57Fe substitution and the one with natural iron. The
central shift and linewidth of the SMS radiation was measured using a single-
line K2Mg57Fe(CN)6 absorber, typically before and after each measurement
point. From those measurements the actual center shift and energy resolution
of the SMS was derived for the data evaluation of the spectra. Typically,
values are 0.395$\pm$0.10 mm/s for the center shift with respect to
$\alpha$-Fe and 0.25$\pm$0.05 mm/s for the energy resolution. The beam from
the SMS was focused using Kirkpatrick-Baez Mirror to the spot size of 16
$\mu$m $\times$ 18 $\mu$m (vertically $\times$ horizontally, FWHM). After a
first measurement made on all samples at the lowest temperature (about 3 K),
ambient pressure and zero magnetic field, for comparison and reproducibility
check, we selected two of them, one with 96% 57Fe substitution (hereafter
named ”enriched iron sample”) and the one with natural iron (hereafter named
”natural iron sample”), for the remaining experiments in temperature and
field. The choice was made on the basis of the smallest width of the line,
corresponding to the best quality (e.g. lower disorder) of the crystal.
Samples with 96% 57Fe substitution experienced mainly line broadening due to
their high effective thickness, while the natural sample gave a resolution
limited broadening, although requiring longer accumulation times.

Figure 9:  (Color online) SMS spectra of LaFeSiH enriched (top panel) and
natural (bottom panel) sample at H=-7 T, T= 6K and P=11.4 GPa. Same symbols
and color code as in Fig. 2 panel a).

In Fig. 7 we show the raw data of Fig. 1, taken with an acquisition time of
about 103 sec.

In Fig. 8 we show a picture of the enriched (S #4) and natural (S #7) samples
loaded in a DAC with 500 $\mathrm{\mu}$m cullet diamonds and stainless steel
gasket was used. The diameter of the gasket hole is 250$\mathrm{\mu}$m, with
70 $\mathrm{\mu}$m thickness chosen to avoid bridging of the single crystal at
the highest applied pressure.

Finally, in Fig. 9 we show the measurements obtained at the highest applied
field and pressure at low temperature. Similar to the measurements at ambient
pressure we do not observe any possible magnetic contribution, as the fitted
hyperfine field 7.14(6) match the applied one within error bar with a very
small residual one.

##  Appendix B Electrical resistivity measurements

Resistance of enriched sample as a function of temperature was measured in the
four contact geometry down to 4.2 K with an excitation current of 100 $\mu$A
using a custom build He4 cryostat. Resistance was measured using several other
excitation currents in order to check the effect of current on the
superconducting transition and normal state resistance behaviour.

Figure 10: (a) Resistance of LaFeSiH single crystal as function of
temperature. (b) Derivative of the resistance as a function of temperature.
(Inset) Zoom for temperature range 5-15 K showing superconducting transition
around 10 K.

Fig. 10 (a) shows the measured resistance of LaFeSiH single crystal as
function of temperature which shows a Residual Ratio Resistivity
$\frac{R(300K)}{R(11K)}$ of $\sim$ 25\. Except the superconducting transition,
no anomaly, in particular no change of slope which could suggest the onset of
an antiferromagnetic state above the superconducting transition temperature is
visible.

To further highlight a possible small change of slope, in Fig. 10, (b), we
plot the derivative of the resistance data as a function of temperature. The
very sharp superconducting transition (T${}_{c}^{90\%}$-T${}_{c}^{10\%}$=0.6K)
is shown in inset. The T${}_{c}^{Onset}\sim$ 10.5 K is slightly above the
average of the whole batch used in Mössbauer experiments. However, no other
signal is detected, contrary to, e.g. LaFeAsO, where a clear maximum in the
derivative of the resistance is detected at the anti-ferromagnetic transition,
as shown in Ref Klauss  _et al._ , 2008. No significant difference were found
in the normal state resistivity with different applied current.

##  Appendix C Magnetization measurements

The magnetic properties of the three different samples were probed in a
Metronique Ingenierie SQUID magnetometer using the standard extraction method:
the deuterated LaFeSiD sample (analysed by neutron powder diffraction and
x-ray diffraction) and both LaFeSiH samples made of natural Fe (for NMR and
Mössbauer experiments) and enriched with Fe57 (for Mössbauer spectroscopy).
Magnetization as a function of applied field (M(H) curves) was measured at
fixed temperature (2 K and 150 K). Magnetization versus temperature (M(T)
curves) was also acquired at different applied fields (10 Oe, 100 Oe and 1
kOe) after cooling the samples down to 2 K in zero field, measuring during
heating (zero field cooled (ZFC) process) and cooling (field cooled (FC) mode)
of the sample.

The superconducting state is clearly identified in both M(T) and M(H) curves
for the different samples, either made from natural Fe or enriched with Fe57.
The M(H) curve (Fig. 11 for LaFeSiD) shows the expected initial negative slope
related to the diamagnetic shielding (at low field, see the linear dependence
in the inset of Fig. 11). From this slope we conclude that superconductivity
is bulk in the sample, with a superconducting volume fraction above 80 % at 2
K for LaFeSiD. Similar or slightly higher values are found for the LaFeSiH
samples. The onset of superconductivity (at 10 Oe) is found in the M(T) curve
at the critical temperature Tc $\sim$= 10 K for LaFeSiD (inset of Fig. 12), at
Tc $\sim$ 8.8 K for the enriched Fe57 LaFeSiH batch (inset of Fig. 13 for the
powder and Fig. 14 for the single crystal), and at Tc = $\sim$ 8.5 K for
LaFeSiH based on natural Fe (M(T) curve not shown). All these values are very
close to the previously reported Tc values for LaFeSiH powder and single
crystal samplesBernardini  _et al._ (2018).

Figure 11:  Magnetic field dependence of LaFeSiD magnetization at T = 2 K up
to 1 kOe. Inset: magnetization at T = 2 K up to 10 Oe. Figure 12:  Temperature
dependence of LaFeSiD magnetization at 10 Oe up to 320 K. Inset: enlarged view
around Tc = 10 $\pm$ 0.2 K. Figure 13: Temperature dependence of isotopic
Fe57-based LaFeSiH (powder) magnetization at 1 kOe up to 315 K. Inset:
enlarged view of M(T) measured at 10 Oe around Tc $\sim$ 8.8 K. Figure 14:
Temperature dependence of isotopic Fe57-based LaFeSiH (single crystal)
magnetization at 10 Oe in the 2-12 K range. Inset: enlarged view of M(T)
measured at 10 Oe around Tc.

In the normal state, no significant anomaly is detected, except the
ferromagnetic transition associated with the minor deuterated La(Fe1-xSix)13Dy
phase which presents a Curie temperature near the maximal reached temperature,
320 KPhejar  _et al._ (2016); Fujieda  _et al._ (2008) (Fig. 12). From the
saturated magnetization of this impurity, one can estimate that our sample
contains less than a weight fraction of $\sim$ 2 % of such phase, which is not
detected in our diffraction experiments. The presence of such ferromagnetic
background could maybe hide less pronounced transition (like an
antiferromagnetic transition) associated with the LaFeSiD main phase. In the
Fe57-based LaFeSiH, this magnetic background is much smaller (see M(T) curves
at 1 kOe, Fig. 12 and Fig. 13) and the signature of potential La(Fe1-xSix)13Hy
impurity is not detected or its Curie temperature is pushed largely above 315
K. Thanks to this smaller background the Curie-Weiss behavior of the main
phase becomes visible and a modelisation by $\chi$ = $\chi$0 \+
C/(T-$\theta$W) gives $\theta$W $\sim$ \- 69 K, highlighting the
antiferromagnetic character of the main magnetic interactions in LaFeSiH. We
notice also that, except superconductivity, no other anomaly associated with
potential structural or electronic/magnetic long range order is directly
detectable in the 2 - 315 K range. Other complementary techniques sensitive to
magnetism have been used in this work to probe such potential local or long
range magnetic order: NPD, Mössbauer and NMR spectroscopies.

##  Appendix D X-ray and Neutron diffraction - comparison

The deuterated sample, LaFeSiD was measured using both neutron- and X-ray
diffraction.

Table 1: The structure of LaFeSiH at T = 300 K and T = 5 K as measured using
XRD.  P4/nmm
---
a = 4.02557(1), c = 8.0227(5) | a = 4.02570(2), c =7.9756(6)
T = 300 K | T = 5 K
Atom | x | y | z | x | y | z
La | 0 | 0.5 | 0.6709(4) | 0 | 0.5 | 0.6722(4)
Fe | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5
Si | 0.5 | 0 | 0.1495(7) | 0.5 | 0 | 0.1502(6)
H | 0 | 0 | 0 | 0 | 0 | 0

In figure 15 we show the XRD patterns collected at various temperatures,
showing no splitting og peaks down to 5 K. In figure 16 we summarize the
refined structural parameters and the structural parameters calculated from
XRD and NPD. Looking first at figure 16.a-c, we show the parameters associated
with the unit cell. Here we see a good agreement between the XRD and NPD data,
although there is a slight offset. We ascribe this offset to the lack of
precision of the neutron wavelength. Then, turning to the atomic position
parameters shown in figure 16.d-f, we see that the z(Si) parameter is not
excellently matched between the two experiments. This can be explained by the
rather low scattering cross-section of Si with neutrons, and we thus find it
suitable to trust the parameters obtained from XRD. For the z(La) parameter we
find a decent match between the two experiments. The thermal parameters, B,
all show low positive values, decreasing as temperature decreases, indicative
of a reliable fit. In figure 16.g-l we show various geometric parameters,
calculated from the refined parameters. We see that the values associated with
Si, show some deviation between the XRD and NPD data, as we could expect given
the above argument, however, the general trends are well matched. Table 1
summarizes the structure at 300 K.

Figure 15: The XRD pattens collected at SOLEIL at different temperatures,
using a Mythen 2 detector. Figure 16: Refined and calculated structural
parameters for LaFeSiD as a function of temperature, measured using XRD and
NPD. a: The unit cell parameter, a. b: The unit cell parameter, b. c: The unit
cell volume. d: The atomic position parameter z(Si). e: the atomic position
parameter z(La). f: The thermal vibration parameters B. For deuterium, the
values are in the range of 1.3 for the NPD data and they were not refined for
the XRD data. g: The Fe-Si height, along the c-axis. h: The La-H(D) height
along the c-axis. i: The La-Si height along the c-axis. j: The Fe-Si bond
length. k: $\alpha$ angle of the Fe-Si tetrahedron. l: The $\beta$ angle of
the Fe-Si tetrahedron.


# Network Inference using Sinusoidal Probing

Robin Delabays1,2 and Melvyn Tyloo3,4 1 Automatic Control Laboratory, Swiss
Federal Institute of Technology, CH-8092 Zurich, Switzerland.
2 Center for Control, Dynamical Systems and Computation, UCSB, Santa Barbara,
CA 93106-5070 USA.
3 School of Engineering, University of Applied Sciences of Western Switzerland
HES-SO, CH-1951 Sion, Switzerland.
4 Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL),
CH-1015 Lausanne, Switzerland.

###### Abstract

The aim of this manuscript is to present a non-invasive method to recover the
network structure of a dynamical system. We propose to use a controlled
probing input and to measure the response of the network, in the spirit of
what is done to determine oscillation modes in large electrical networks. For
a large class of dynamical systems, we show that this approach is analytically
tractable and we confirm our findings by numerical simulations of networks of
Kuramoto oscillators. Our approach also allows us to determine the number of
agents in the network by probing and measuring a single one of them.

###### keywords:

Networks, inference processes, Jacobian matrices, input signals, low
frequencies.

††thanks: This work has been supported by the Swiss National Science
Foundation under grants 200020_182050 and P400P2_194359.

##  1 Introduction

Complex networks are the medium for interactions in many natural and man-made
systems. Such realizations range from the scale of people exchanging opinions
on a social networks and power transmission on electrical grids to interacting
molecules in chemical reactions and pacemaker cells [Boccaletti et al. (2006);
Newman (2018)]. The way individual elements are coupled together primarily
impacts the overall dynamics of network-coupled systems. However, in many
cases, characteristics of the interaction network are not known exactly, or
even not known at all. Inference techniques to uncover coupling between
individual units and even units with themselves are therefore highly
desirable. We distinguish mainly two types of such methods, (i) one can
observe a dynamical system subjected to uncontrolled operational condition; or
(ii) one can directly disturb the system and observe its reaction, which is
what is done in this manuscript. In real applications, one has to be careful
on the nature of the method. Introducing a disturbance into the system can
have dramatic effect on its operation. For instance, Furutani et al. (2019)
propose a method based on resonance of the network when subject to periodic
disturbance, and Timme (2007) suggests to drive the system away from its
operating state. Such (potentially invasive) methods could alter the
operational state of the system.

To avoid a strong impact on the system, one can adopt the strategy of simply
observing the system in its normal operation. This type of approach lead to
successful and elegant results, e.g., leveraging the response of the system to
external noise [Ren et al. (2010); Tyloo et al. (2020)]. But as there is no
free lunch, these methods come with assumptions on the noise characteristics
that are not necessarily met in general, namely on correlation time and
uniformity over the system. Observing the relaxation of a system to its steady
state, Mauroy and Hendrickx (2017) extract its spectral moments, but do not
directly reconstruct the network. Furthermore, their method performs exactly
for linear dynamics, but relies on Dynamical Mode Decomposition for nonlinear
systems, which might require a very large number of measurements for an
accurate estimation.

Half way between intrusive disturbances and passive observation, we will
consider the injection of a small probing signal, which we will qualify as
”non-invasive”, in order to leave the operational state as unaffected as
possible in the spirit of Pierre et al. (2010). To this day, most of such
techniques, applied to large power grids, relied of very few measurement
points and aimed only at identifying resonance modes of the network, and not
the whole network structure [Dosiek et al. (2013)]. In this manuscript, we
propose such a non-invasive inference technique. It relies on rather mild
assumptions on the nature of the interaction between agents of the network,
and does not need to know it, on the contrary to other methods [e.g., Yu et
al. (2006)]. Our method applies generically to any network and does not need
any knowledge of its charateristics, as for instance in Yeung et al. (2002).
By adding a controlled sinusoidal input at single nodes, referred to as
_probing signal_ , we are able to reconstruct the interaction network by
measuring the response to the probing at the other nodes of the network. The
same approach allows us to determine the number of nodes in the network by
probing and measuring the response of the network at a single node. This
improves significantly on previous measurement-based methods to determine the
number of nodes in a network [Haehne et al. (2019)]. In contrast to more
exhaustive works [e.g., Materassi and Innocenti (2010); Dankers et al. (2012)
and other publication of the same authors], where the authors aim at
identifying both network structure, transfer functions, and internal node
dynamics, our method focusses on the interaction network only, avoiding to
rely on an error minimization, which cannot always be computed in closed form.

##  2 Preliminaries

Let us consider a general network of $n$ coupled agents with first-order
dynamics and diffusive coupling

$\displaystyle\dot{y}_{i}$
$\displaystyle=\omega_{i}-\sum_{j}a_{ij}f_{ij}(y_{i}-y_{j})+\xi_{i}\,,$
$\displaystyle i$ $\displaystyle=1,...,n\,,$ (1)

where $y_{i}\in\mathbb{R}$ is the time-varying value of the $i$th agent,
living on a one-dimensional manifold parametrized by $\mathbb{R}$,
$\omega_{i}\in\mathbb{R}$ is the natural driving term of agent $i$, and
$\xi_{i}$ will be used as an input to the system. Two agents $i$ and $j$ are
interacting if a link between them exist in the interaction network, i.e., if
and only if the corresponding term of the adjacency matrix $a_{ij}=1$. The
interaction function between $i$ and $j$ is an odd, differentiable function
$f_{ij}\colon\mathbb{R}\to\mathbb{R}$, and the coupling is symmetric, i.e.,
$f_{ij}=f_{ji}$. We also consider an attractive coupling, i.e., $\partial
f_{ij}/\partial y>0$ in an interval around $y=0$.

If a fixed point $\bm{y}^{*}\in{\cal M}^{n}$ exists, one can linearize Eq. (1)
around it, which yields, for a small deviation $\bm{x}=\bm{y}-\bm{y}^{*}$, to
approximate the dynamics

$\displaystyle\dot{\bm{x}}$ $\displaystyle=-{\cal
J}_{\bm{f}}(\bm{y}^{*})\bm{x}+\bm{\xi}\,,$ (2)

where we use the Jacobian matrix of Eq. (1),

$\displaystyle{\cal J}_{\bm{f},ij}(\bm{y}^{*})$
$\displaystyle=a_{ij}\frac{\partial}{\partial
y}f_{ij}(y_{i}^{*}-y_{j}^{*})\,.$ (3)

One can verify that the oddness of the interaction ($f_{ij}$ odd) and the
symmetry of the dynamics ($f_{ij}=f_{ji}$) implies that the Jacobian ${\cal
J}_{\bm{f}}$ is a weighted Laplacian matrix of the interaction graph. It is
then real symmetric, which implies that it has real eigenvalues,
$\lambda_{1}\leq...\leq\lambda_{n}$, and its eigenvectors,
$\bm{u}_{1},...,\bm{u}_{n}$, form an orthonormal basis of $\mathbb{R}^{n}$.
From now on, we will focus on stable fixed points of Eq. (1), implying that
the eigenvalues are nonnegative. Also, diffusive couplings imply that one
eigenvalue vanishes, $\lambda_{1}=0$, with associated eigenvector
$\bm{u}_{1}=n^{-1/2}(1,...,1)^{\top}$.

Remark. For other types of couplings where $\lambda_{1}>0$, the inference
method proposed in Sec. 6 is even simpler than our case, we do not detail it.
However, the result of Sec. 5 requires the zero mode and cannot be extended
straighforwardly to nondiffusive couplings.

Equation (2) is then solved by expanding the deviation $\bm{x}$ over the
eigenvectors $\bm{u}_{\alpha}$ of ${\cal J}_{\bm{f}}$, i.e.,
$x_{i}(t)=\sum_{\alpha}c_{\alpha}(t)u_{\alpha,i}$, with

$\displaystyle c_{\alpha}(t)$
$\displaystyle=e^{-\lambda_{\alpha}t}\int_{0}^{t}e^{\lambda_{\alpha}t^{\prime}}\bm{u}_{\alpha}^{\top}\bm{\xi}(t^{\prime}){\rm
d}t^{\prime}\,.$ (4)

Details can be found, e.g., in Tyloo et al. (2018).

In order to get numerical confirmation of our results, we will apply them to
the Kuramoto model on three different interaction graphs. The first one is a
representation of the UK electrical grid (see inset in Fig. 3) with $n=120$
vertices and $m=165$ edges. The second one is a realization of an Erdös-Renyi
graph with $n=120$ vertices and $m=329$ edges. The third one is a small-world
graph realized according to the Watts-Strogatz process [Watts and Strogatz
(1998)], with $n=120$ vertices and $m=242$ edges.

Figure 1: Fraction of responding agents vs. excitation frequency $\omega$ for
the three networks UK (left panel), Erdös-Renyi (middle panel), and Small-
World (right panel). Vertical dashed lines correspond to $\lambda_{2}$ and
$\lambda_{n}$ .

##  3 Probing

In order to determine oscillation modes in large electrical networks, one
method is to apply a probing signals at some points of the network and measure
the system’s response at other points [e.g., Pierre et al. (2010)]. The
probing is typically a sinusoidal signal with controlled amplitude and
frequency.

In the same spirit, we propose here to inject a sinusoidal signal at agent $i$
and to measure its impact at agent $j$. Let

$\displaystyle\xi_{i}(t)$ $\displaystyle=a_{0}\sin(\omega_{0}t)\,,$ (5)

be the probing signal at agent $i$. We do not inject a probing signal at other
nodes. Note that any signal shape could do the job, but the advantage of sine
is that its amplitude and frequency are easily identifiable.

To guarantee a minimal impact on the operation of the system, we need the
amplitude $a_{0}$ to be sufficiently small. However, in most applications, the
system under investigation will be subject to noise, and in order to be
detectable, the probing amplitude should not be too small neither. Despite
these contradicting requirements, we argue that, under our assumptions, an
appropriate amplitude, satisfying both requirement simultaneously, can be
chosen. Indeed, even though the system is subject to noise, we assume it is
close to a steady state. If the noise amplitude was too large, it would push
the system far away from its fixed point and it could not be considered as in
(or close to) steady state. Under our assumptions, there is then a margin
between the noise amplitude and the size of the basin of attraction of the
steady state. The probing amplitude needs to be chosen in this margin.
Determining the amplitude of the noise is rather straightforward from
measurements. However, estimating the maximal tolerable disturbance magnitude
preserving stability might represent a challenge and is beyong the scope of
this manuscript [Menck et al. (2013)].

Also, the probing frequency $\omega_{0}$ needs to be kept small. Keeping a
small probing frequency guarantees that the system can adapt to the input and
follow the probing signal. More precisely, a probing frequency can be
qualified as small as long as it is smaller than the smallest eigenvalue of
the Jacobian matrix ${\cal J}_{\bm{f}}$ in absolute value.

Introducing Eq. (5) into Eq. (4), and recombining the eigenmodes yields the
following response measured at agent $j$,

$\displaystyle\begin{split}x_{j}^{i}(t)&=\sum_{\alpha}\frac{u_{\alpha,i}u_{\alpha,j}a_{0}}{\lambda_{\alpha}^{2}+\omega_{0}^{2}}\\\
&\times\left[\lambda_{\alpha}\sin(\omega_{0}t)+\omega_{0}e^{-\lambda_{\alpha}t}-\omega_{0}\cos(\omega_{0}t)\right]\,.\end{split}$
(6)

To explicitly obtain an expression involving the Jacobian matrix, one should
consider the long time limit $\lambda_{\alpha}t\gg 1$ with the asymptotic
$\omega_{0}\ll\lambda_{\alpha}$ that yields,

$\displaystyle\begin{split}x_{j}^{i}(t)&=\left[\sum_{\alpha\geq
2}\frac{u_{\alpha,i}u_{\alpha,j}}{\lambda_{\alpha}}\right]a_{0}\sin{(\omega_{0}t)}+\frac{u_{1,i}^{2}a_{0}}{\omega_{0}}[1-\cos(\omega_{0}t)]\\\
&={\cal
J}_{\bm{f},ij}^{\dagger}a_{0}\sin{(\omega_{0}t)}+\frac{a_{0}}{n\omega_{0}}[1-\cos{(\omega_{0}t)]}\,,\end{split}$
(7)

where the $\dagger$ denotes the Moore-Penrose pseudo-inverse. We will say that
a pair of agents $(i,j)$ can be probed if we have the ability to inject a
probing signal at one of them and to measure the response at the other.

Note that even though one usually has access to the dynamical variable $y_{i}$
instead of $x_{i}$, the latter can be recovered by comparing the behavior of
$y_{i}$ before and after the introduction of the probing,

$\displaystyle x_{i}(t)$ $\displaystyle=y_{i}(t)-y_{i}^{*}\,.$ (8)

##  4 Spectrum range estimation

One condition in order to derive Eq. (7) is that the probing frequency
$\omega_{0}$ is much smaller than all eigenvalues of the Jacobian matrix. From
a practical point of view, however, the Jacobian being unknown, one cannot
guarantee a priori to choose a frequency sufficiently small.

Using the fact that we can probe and measure the system, we propose here a
method to infer the range of the spectrum of the Jacobian Eq. (3). It can be
inferred by tuning the signal’s frequency $\omega_{0}$ and observing how many
agents, $n_{\rm act}$, responded to the signal. Indeed, when
$\omega_{0}\gg\lambda_{\alpha}$, one verifies in Eq. (6) that
$x_{j}^{i}=O(\omega_{0}^{-1})$, which vanishes for large $\omega_{0}$, meaning
that the signal stays local and do not spread across the network. For
$\omega_{0}\ll\lambda_{\alpha}$, Eq. (7) shows that for sufficiently small
$\omega_{0}$,

$\displaystyle x_{j}^{i}(t)$
$\displaystyle\approx\frac{a_{0}}{n\omega_{0}}[1-\cos(\omega_{0}t)]\,,$ (9)

independently of $i$ and $j$. All agents in the network then respond together
to the signal (i.e., adiabatic shift in the parameters).

Injecting a probing signal and varying its frequency, one can then estimate
the range of the Jacobian’s spectrum, which is delimited by the frequencies
where probing propagates to the whole network and where it does not propagate
at all respectively. This is illustrated in Fig. 1, where each cross
corresponds to the fraction of agents that responded to the probing at a
single agent. One observes that the boundaries (vertical dashed lines) of the
spectrum of ${\cal J}_{\bm{f}}$ are roughly estimated by the transition from
the probing signal staying local (large $\omega_{0}$) to spreading over the
whole network (small $\omega_{0}$).

This gives at least an order of magnitude of the spectrum range, and hence of
what is a ”small” probing frequency.

##  5 Number of agents

The number of agents in a coupled system is one of its primal properties.
However, there are many physical examples where this number is not known
exactly [Su et al. (2012), Haehne et al. (2019)] In this section, we introduce
a method that allows to accurately determine the number of agents in any
system governed by the dynamics of Eq. (2). Moreover, it requires to probe and
measure a single node, which makes the method very efficient. Let us inject a
probing signal, Eq. (5) at node $i$, with $\omega_{0}\ll\lambda_{\alpha}$.
Then for $\omega_{0}$ sufficiently small one has,

$\displaystyle x_{i}^{\rm max}=\max_{t}|x_{i}(t)|$
$\displaystyle\approx\frac{2a_{0}}{n\omega_{0}}\,,$ (10)

from which one obtains an estimate for the number of nodes as,

$\displaystyle\hat{n}=\frac{2a_{0}}{x_{i}^{\rm max}\omega_{0}}\,.$ (11)

Note that we take the maximum to have a better accuracy in the estimation.
However one can choose a particular time step $t$, keeping in mind that $t$
too short leads to vanishing values for Eq. (7). We check the validity of the
estimation of Eq. (11) in Fig. 2. Each cross corresponds to $\hat{n}$ obtained
from a single node probing and measurement. One clearly sees that for
$\omega_{0}$ small enough compared to $\lambda_{2}$, the estimated number of
agents precisely matches the real one.

Figure 2: Estimation of the number of agents from a single probing/measurement
for the three networks UK, Erdös-Renyi, and Small-World. All of them have
$n=120$ vertices. Left panel, UK: $\omega_{0}/\lambda_{2}=0.02$ (orange),
$\omega_{0}/\lambda_{2}=0.01$ (blue); Middle panel, ER:
$\omega_{0}/\lambda_{2}=0.0025$ (orange), $\omega_{0}/\lambda_{2}=0.00126$
(blue); Right panel, SW: $\omega_{0}/\lambda_{2}=0.01$ (orange),
$\omega_{0}/\lambda_{2}=0.005$ (blue) . Some crosses are slightly higher than
$120$ because of constant shifts in trajectories, as described in Sec. 6.

Since the initial submission of this manuscript, this result has been
developed and thoroughly discussed in Tyloo and Delabays (2020).

##  6 Network inference

Assume now that each of the $[n(n-1)]/2$ pairs $(i,j)$, $1\leq i<j\leq n$, can
be probed. For each pair $(i,j)$, we can measure either $\hat{x}_{j}^{i}(t)$
or $\hat{x}_{i}^{j}(t)$. According to Eq. (7) and by symmetry of ${\cal
J}_{\bm{f}}$, these two trajectories should be the same, at least for $t$
sufficiently large, in order for the influence of initial conditions to
vanish.

We will use the measured value $\hat{x}_{j}^{i}(t)$ at large $t$ to estimate
${\cal J}_{\bm{f}}^{\dagger}$ via Eq. (7). To do this, we need to remember
that the dynamics of Eq. (1) are invariant under a constant shift of all
variables. This implies that two trajectories whose initial conditions differ
only by a constant shift of all variables are exactly parallel, i.e., the
constant shift is preserved for all time. This means that, depending on the
initial conditions (unknown a priori), the measured trajectory
$\hat{x}_{j}^{i}(t)$ might be shifted with respect to the predicted trajectory
$x_{j}^{i}(t)$ of Eq. (7). Denoting this constant shift as
$c\coloneqq\hat{x}_{j}^{i}(t)-x_{j}^{i}(t)$, one can rearrange Eq. (7) as

$\displaystyle\begin{split}\hat{\cal J}^{\dagger}_{\bm{f},ij}&\coloneqq{\cal
J}^{\dagger}_{\bm{f},ij}+\frac{c}{a_{0}\sin(\omega_{0}t)}\\\
&=\left\\{\hat{x}_{j}^{i}(t)-\frac{a_{0}}{n\omega_{0}}\left[1-\cos(\omega_{0}t)\right]\right\\}/\left[a_{0}\sin(\omega_{0}t)\right]\,.\end{split}$
(12)

Whereas we cannot determine the value of $c$, this is not an issue, as we show
now. Indeed, the constant vector $\bm{u}_{1}=n^{-1/2}(1,...,1)^{\top}$ is an
eigenvector of ${\cal J}_{\bm{f}}$, and then an eigenvector of ${\cal
J}_{\bm{f}}^{\dagger}$ as well. As the eigenbasis of ${\cal J}_{\bm{f}}$ (and
${\cal J}_{\bm{f}}^{\dagger}$) is orthonormal, adding a constant value to each
componenent of ${\cal J}_{\bm{f}}^{\dagger}$ does not modifies its eigenbasis,
and modifies only one of its eigenvalues, $\lambda_{1}$. Diagonalizing
$\hat{\cal J}_{\bm{f}}^{\dagger}$, it is then straightforward to replace its
eigenvalue associated to $\bm{u}_{1}$ by $0$ (as it should be for the exact
${\cal J}_{\bm{f}}$ and ${\cal J}^{\dagger}_{\bm{f}}$) and to invert all other
eigenvalues to recover ${\cal J}_{\bm{f}}$. Remark that in order to avoid
singularity in Eq. (12), one should choose a time step $t$ such that
$\sin(\omega_{0}t)$ is sufficiently different from zero.

The outcome of the procedure proposed above is illustrated in Fig. 3. Each dot
corresponds to one element of the $120\times 120$ Jacobian matrix for the
Kuramoto model on our selected graphs. One can see that the matrix is
reconstructed with very high accuracy. In Fig. 4, we show the accuracy of the
Jacobian estimate with respect to the frequency of the probing, normalized by
$\lambda_{2}$. The accuracy of the estimate is measured as the Frobenius norm
of the difference between the real Jacobian matrix and its estimate. We see
that the accuracy is very good, even for probing frequencies close to
$\lambda_{2}$.

Figure 3: Inferred vs. real values of the elements of the Jacobian matrix
${\cal J}_{\bm{f}}$ for the three networks UK (left panel), Erdös-Renyi
(middle panel), and Small-World (right panel). Figure 4: Relative error in the
estimate of the Jacobian matrix $\hat{\cal J}_{\bm{f}}$ with respect to the
frequency of the probing signal, normalized by the smallest eigenvalue of the
Jacobian, for the three networks UK (blue), Erdös-Renyi (orange), and Small-
World (green). The relative error is computed as the Frobenius norm of the
error of the Jacobian matrix normalized by the Frobenius norm of the real
matrix. One sees that for probing frequencies smaller that $10\%$ of
$\lambda_{2}$, the estimate is very accurate.

##  7 Outlook

We showed that based on a sinusoidal probing signal, injected at a node of a
networked dynamical system while measuring the response of the network allows
to:

  * •

Estimate the range of the spectrum of the Jacobian matrix of the system around
its current fixed state;

  * •

Estimate the number of agents in the system very efficiently: single node
measurement;

  * •

Recover the network structure of the system with high fidelity.

The main advantages of our method are that it is non-invasive (does not
disturb the system far from its operating state), it applies to a large set of
coupling functions, whereas it requires to probe the system for a sufficiently
long time, it only needs measurement at one time step. Its main drawback is
that, in order to recover the network using sinusoidal probing, one needs to
be able to probe any pair of nodes in the network. The subsequent work would
then investigate to what extent and with what confidence the network structure
can be inferred if one has only probed a subset of the nodes.

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