# Nematic state of the FeSe superconductor

Sahana Rößler<EMAIL_ADDRESS>Max Planck Institute for Chemical Physics
of Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany  Mauro Coduri
University of Pavia, Chemistry Department, via Taramelli 16, 27100, Pavia,
Italy  Alexander A. Tsirlin  Experimental Physics VI, Center for Electronic
Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany
Clemens Ritter  Institut Laue-Langevin, 71 Avenue des Martyrs, CS20156, 38042
Grenoble Cédex 9, France  Gabriel Cuello  Institut Laue-Langevin, 71 Avenue
des Martyrs, CS20156, 38042 Grenoble Cédex 9, France  Cevriye Koz  Max Planck
Institute for Chemical Physics of Solids, Nöthnitzer Straße 40, 01187 Dresden,
Germany  Liudmila Muzica  Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Straße 40, 01187 Dresden, Germany  Ulrich Schwarz  Max Planck
Institute for Chemical Physics of Solids, Nöthnitzer Straße 40, 01187 Dresden,
Germany  Ulrich K. Rößler  IFW Dresden, Helmholtzstraße 20, 01069 Dresden,
Germany  Steffen Wirth  Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Straße 40, 01187 Dresden, Germany  Marco Scavini
<EMAIL_ADDRESS>Dipartimento di Chimica, Universita degli Studi di
Milano, Via Golgi 19, 20133 Milano, Italy

###### Abstract

We study the crystal structure of the tetragonal iron selenide FeSe and its
nematic phase transition to the low-temperature orthorhombic structure using
synchrotron x-ray and neutron scattering analyzed in both real and reciprocal
space. We show that in the local structure the orthorhombic distortion
associated with the electronically driven nematic order is more pronounced at
short length scales. It also survives up to temperatures above 90 K where
reciprocal-space analysis suggests tetragonal symmetry. Additionally, the
real-space pair distribution function analysis of the synchrotron x-ray
diffraction data reveals a tiny broadening of the peaks corresponding to the
nearest Fe–Fe, Fe–Se, and the next-nearest Fe–Se bond distances as well as the
tetrahedral torsion angles at a short length scale of 20 Å. This broadening
appears below 20 K and is attributed to a pseudogap. However, we did not
observe any further reduction in local symmetry below orthorhombic down to 3
K. Our results suggest that the superconducting gap anisotropy in FeSe is not
associated with any symmetry-lowering short-range structural correlations.

††preprint: Regular article

##  I Introduction

The discovery of superconductivity in FeSe Hsu2008  with a critical
temperature $T_{c}\approx 8\,$K prompted intense research in this remarkable
family of materials Boh2018 ; Kre2020 ; Shi2020 . FeSe, the structurally
simplest among the iron-based superconductors, was considered to be an ideal
candidate for studying the mechanism of unconventional multiband
superconductivity. In spite of the expectations, more than a decade of
research proved FeSe as a rather unusual superconductor with several
fascinating properties in the normal state, i.e., above $T_{c}$.

A structural transition from the parent tetragonal ($P4/nmm$, No. 129) crystal
structure to the orthorhombic $Cmma$ (new symbol $Cmme$, No. 67) space group
occurring at a temperature $T_{s}\approx 90$ K is considered to be driven by
electronic degrees of freedom, and hence referred to as a nematic transition.
However, the energy scale of this nematic order ranges from 10–50 meV, as
determined from angle-resolved photoemission experiments (ARPES), and is
controversially discussed in the literatureFed2016 ; Col2018 ; Yi2019 . At
ambient pressure, the nematic transition is not accompanied by a long-range
magnetic order Mc2009a . Recent reports on pair distribution function (PDF)
analysis of scattering data have shown that the short-range orthorhombic
distortions are already present at room temperature with a growing domain size
upon decreasing temperature Koch2019 ; Benj2019 ; Kon2019 ; Wang2020 .

Even though the primary order parameter of the phase transition at 90 K is
considered as nematic and of electronic origin, the exact nature of the order
parameter remains unknownFer2014 . Several possible order parameters have been
suggested, which include spin Wang2016a , charge Mass2016 , or orbital
Baek2014  degrees of freedom, antiferroquadrupolar order Yu2015 , stripe
quadrupolar order Zha2017 , and collective modes such as a Pomeranchuk
instability Mass2016 ; Kle2018  of the Fermi surface. Indeed, in a quantum
material such as FeSe, the order parameters cannot be considered as
independent and competing. Instead, the order parameters are composite and
intertwined as suggested by recent theoretical studies Fer2019 .

Below $T_{s}$, FeSe displays two more anomalies Ros2015 ; Ros2018 ; Su2016 ;
Ter2016 ; Imai2009 ; Gri2018 ; Kas2014 ; Ghini2021 ; Kasa2016  before entering
into the superconducting phase at 8 K. The anomaly at $T^{*}\approx 75$ K was
identified in nuclear magnetic resonance (NMR) Baek2014 ; Imai2009 , muon spin
relaxation ($\mu$SR) Gri2018 , Kohler’s scaling behavior of magnetoresistance,
and Hall effect measurements Ros2015 ; Ros2018 . Since all these techniques
probe the underlying magnetic response of the sample, the temperature $T^{*}$
is considered as an onset temperature of anisotropic spin-fluctuations. Upon
further lowering of temperature to $T^{**}\approx 30-20$ K, an inflection was
observed in Hall, Seebeck, and Nernst coefficients Ros2015 ; Kas2014 ;
Kasa2016 . The Kohler’s scaling was found to be reestablished below $T^{**}$
Ros2015 ; Ros2018 ; Su2016 ; Ter2016 . Although the signature of $T^{**}$ was
identified in many different techniques, its interpretation differs in
literature. Several authors propose giant superconducting fluctuations
Kasa2016 ; Sol2020  or a pseudogap related to preformed Cooper pairs assuming
that the system is in the cross-over regime between a weak-coupling Bardeen-
Cooper-Schrieffer (BCS) and strong-coupling Bose-Einstein condensate (BEC)
limits Kas2014 ; Kasa2016 . A breakdown of quantum critical fluctuations owing
to a temperature-induced Lifshitz transition was proposed based on the results
of $\mu$SR measurements Gri2018 . A study based on transmission electron
microscopy (TEM) suggested a possible crystal symmetry lowering below 20
KMc2009a .

For the mechanism of superconductivity in Fe-based materials, several
different pairing scenarios are considered Kre2020 ; Maz2008 ; Kon2010 ;
Islam2021 ; Fer2022 . Experimental studies based on scanning tunneling
microscopy/spectroscopy (STM/S) and ARPES on FeSe have reported highly
anisotropic superconducting gaps with either deep gap minima Lin2016 ; Spr2017
; Kush2018  or existence of nodes Kas2014 ; Hashi2018 . Detailed mappings of
the Fermi surfaces have found a $\Gamma$–centered hole pocket and an
$X$–centered electron pocket with the gap minima along the $k_{y}$ axis for
the $\Gamma$ pocket and along the $k_{x}$ axis for the $X$ pocket,
respectively Spr2017 ; Kas2014 . This strong anisotropy in the superconducting
gap symmetry has been attributed to orbital selective interactions Nic2013 ;
Liu2018 ; Steff2021 ; Pfau2021 ; Bart2021 . Moreover, inelastic neutron
scattering studies have identified a prominent role of spin-orbit coupling
(SOC) in FeSe Ma2017 , which provides anisotropy to the spin channel. Through
the SOC, the spin degrees of freedom couple to the lattice Lav2002 ; Ross2019
.

With the motivation to probe the couplings of the electronic degrees of
freedom to the crystal structure, we performed synchrotron powder x-ray
diffraction and powered neutron scattering experiments in the tetragonal,
nematic, and superconducting phases of FeSe. The powder diffraction
experiments allow the highest resolution in order to distinguish the possible
symmetry lowering below orthorhombic and can be directly converted into pair
distribution function as the best tool for probing the local structure. These
experiments enable us to identify if any incipient structural ordering modes
are associated with the observed anomalies in the physical properties at
different temperatures.

Figure 1: A comparison of parameters obtained from the reciprocal-space
refinement of neutron (D2B) and x-ray (XRD) diffraction data, as well as real-
space analysis of $G(r)$ in different $r$-ranges (using both x-ray and
neutron, assigned as XRD and N, respectively in the legend). (a) Ratios of
lattice parameter $b$/$a$ and (b)-(e) anisotropic displacement parameters
$U_{11}$, $U_{22}$ and $U_{33}$, obtained from both reciprocal-space
refinements of D2B measurements and real-space Rietveld analysis of D4c $G(r)$
functions for different $r$ ranges. In the $G(r)$ refinements, we fixed
$U_{11}$ = $U_{22}$ for both Fe and Se. Legend in (a) applies to all panels.
The error bars for the D2B and XRD data shown in (a) are smaller than the
symbol size.

##  II Experimental

Polycrystalline FeSe samples were synthesized by following the procedure
described in Ref. [McQ, ]. The samples were characterized using laboratory
x-ray diffraction.

The synchrotron x-ray diffraction was carried out at the beamline ID22
(wavelength $\lambda=0.354144(6)\,\mathrm{\AA}$) of the ESRF (European
Synchrotron Radiation Facility, Grenoble) using a special He-flow cryostat or
a cold N2–gas blower (cryostream) adapted to the diffraction setup. At ID22, 9
scintillation detectors preceded by Si analyzer crystals were used. This
allowed an extremely high resolution but also caused long counting times in
PDF measurements. The powder sample was loaded into a 0.5 mm glass capillary
and closed with a wax to allow a good thermal contact of the sample with an
exchange gas. Here we would like to note that the very high photon flux of the
ID22 beamline causes local heating of the sample. Hence, the x-ray beam was
attenuated by about 70 % for our measurements. Further attenuation of the beam
did not change the lattice parameters or the temperature of the tetragonal-
orthorhombic transition indicating a stable local sample temperature.

The x-ray diffraction data were collected at temperatures from 5 to 150 K, and
in the $2\theta$ range $2^{\circ}\leq 2\theta\leq 90^{\circ}$ using the
cryostat due to system constraints, and in the range $2^{\circ}\leq
2\theta\leq 120^{\circ}$ for measurements using the cryostream. In both cases
the scanning rate was $4^{\circ}$/min. For PDF analysis, the data were
collected by summing several scans for a total counting time of 385 min per
measured temperature to achieve the necessary data quality for analysis.

The neutron scattering experiments were performed at ILL (Institut Laue
Langevin, Grenoble). For these measurements, FeSe powder samples were
synthesized in several batches and about 5 g of impurity freeKoz2014  FeSe
powder was selected. The sample was filled in a vanadium container.

The neutron diffraction experiments were performed in two different
instruments. For the reciprocal-space structure analysis (Rietveld refinement)
the data were collected on the high resolution powder diffractometer D2B
($\lambda=1.596\,\mathrm{\AA}$) in the temperature range 3–90 K in the heating
cycle (ILL D2B data Ref. [RefD4data, ].) For real-space PDF analysis, the
diffraction patterns were collected at the D4c instrument Fischer2002
($\lambda=0.4959\,\mathrm{\AA}$) at temperatures 150 and 7 K upon cooling and
in the temperature range 3-150 K plus 298 K, upon heating (see ILL D4c data
Ref. [RefD4data, ]). However, since no hysteresis was found in the structural
data, we present here only the data collected in the heating cycle.

Reciprocal-space refinement of the x-ray and neutron diffraction data was
performed using the Rietveld method implemented in GSASGsas  and JANA2006Jana
. Moderate effects of preferred orientation caused by the large anisotropy of
the crystal structure were accounted for by the March-Dollase formalism
Mar1932 ; Doll1986 .

The X-ray PDF curves G(r) were computed using the PDFgetX3 program PDFGetX3 .
The neutron D4c data reduction was performed using standard instrument-
dependent software available at the ILL. The CORRECT program Howe1996  was
then used to conduct the necessary corrections (see, for example ref.
[Fischer2006, ]) to the diffraction data for sample and container attenuation,
multiple-scattering, and normalization of the diffraction intensity to an
absolute scale using a vanadium standard. The resulting $S(Q)$ data were then
Fourier transformed to a PDF using the standard sine integral.

##  III Results

Here, we would first like to outline a few advantages of using both
synchrotron x-ray and neutron scattering experiments for structural analysis.
For the reciprocal-space structure analysis, neutron data are often preferred,
because the reflection intensity does not go down with
$Q=4\pi\mathrm{sin\theta}/\lambda$, so that we can observe a higher number of
intense reflections and use them in the structure refinement. Moreover,
neutrons offer a broader and more symmetric peak shape, whereas x-rays may
show different sorts of anisotropic reflection broadening due to the sampleﬂs
microstructure. Such anisotropic broadening is undesirable for the structure
refinement.

For the real-space structure analysis, the x-rays and neutrons provide
complementary information. With the instruments at the ID22, we reached larger
$Q_{max}$ values using both the cryostat ($\approx\,25\,\mathrm{\AA}^{-1}$)
and the cryostream ($\approx\,29-30\,\mathrm{\AA}^{-1}$). Additionally, the
high $Q$ resolution of ID22 allows to analyze the data up to several
nanometersDej2018 . In the case of neutrons, the usable neutron wavelengths at
the D4c diffractometer are 0.3, 0.5, and 0.7 $\mathrm{\AA}$. However, the
neutron flux is too low at $\lambda\approx 0.3\,\mathrm{\AA}$ to get good data
for the PDF analysis in a reasonable time. Therefore we used the next shortest
available neutron wavelength $\lambda\approx 0.5\,\mathrm{\AA}$, which gives a
$Q_{max}\approx\,23.6\,\mathrm{\AA}^{-1}$. Further, the x-ray measurements are
complementary to neutrons in the sense that, for neutrons the weight of $G(r)$
is larger for Fe than Se. For the x-rays, the opposite is true. Thus, by
combining the two techniques, we are able to maximize the information about
the low temperature structure of FeSe.

Analysis of the interatomic distances made use of the reduced PDF, $G(r)$,
which is the sine Fourier transform of the experimental total scattering
function, $S(Q)$, defined as Egami2003 ; All2012 :

$\displaystyle G(r)$ $\displaystyle=$ $\displaystyle 4\pi
r\,[\rho(r)-\rho_{0}]$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\,\int_{Q_{\textrm{min}}}^{Q_{\textrm{max}}}\,Q[S(Q)-1]\sin(Q.r)\;dQ,$

where $\rho(r)$ is the atomic number density function and indicates the
probability of finding an atom at a distance $r$ from another atom while
$\rho_{0}$ is the average atom number density. The $G(r)$ function measures
deviations from the average atomic density. Provided that the scattering
lengths of two atoms involved in a certain $G(r)$ peak take the same sign, a
positive peak in the $G(r)$ pattern indicates a range of $r$ values whereby
the probability of finding interatomic vectors is greater than that determined
by the number density, while the opposite holds for negative $G(r)$ values.

Figure 2: Schematic representations of FeSe unit cells in (a) tetragonal and
(b) orthorhombic structures. (c) and (d), $ab$ plane of the tetragonal and
orthorhombic structures, respectively. The lattice parameters $a$, $b$, $c$,
inter-atomic distances, and the torsion angle $\phi$ are marked in the
figures. Here, notations $d_{\mathrm{FeSe}}$ and $d_{\mathrm{FeFe}}$ represent
nearest neighbor, NNN $d_{\mathrm{FeSe}}$ next-nearest neighbor inter-atomic
distances, respectively. To distinguish three different Se–Se distances
$d_{\mathrm{SeSe}}$, numbers (1), (2), and (3) are used, which represent
nearest, next nearest, and next-next-nearest neighbor Se–Se, respectively.
Figure 3: (a) and (b) Low $r$ range of $G(r)$ calculated at different
temperatures using synchrotron x-ray diffraction. Here, the notations AFe-Se,
BFe-Fe, and C${}^{\prime}_{\mathrm{Se-Se}}$ represent nearest neighbor, CFe-Se
and DSe-Se next-nearest neighbor, and EFe-Se next-next-nearest neighbor inter-
atomic distances, respectively. (c) The peak positions obtained by the direct
analysis of $G(r)$ functions (symbols) and by the reciprocal-space Rietveld
analysis (dashed lines). (d) The temperature dependence of full-width half-
maxima (FWHM) values obtained from the $G(r)$ peaks displayed in (a) and (b).
The error bars in (c) are smaller than the symbol size.

With a focus on investigating local structural distortions associated with the
anomalies observed at temperatures $T^{*}$ and $T^{**}$, we compare the
parameters obtained from the analysis of the neutron and x-ray diffraction
data for the orthorhombic crystal structure with $Cmma$ symmetry using the so
called real-space Rietveld method PDFGui . In Fig. 1(a), the ratio of lattice
parameters $b$/$a$ obtained from the reciprocal-space Rietveld analysis
(average value of $b$/$a$) and $G(r)$ analysis for different $r$ ranges are
plotted. For the x-ray diffraction data, only the average $b$/$a$ ratio and
that for the $r$ range 2-7 Å is presented for clarity. Compared to the average
values, the $b$/$a$ ratio increases for refinements focusing on the short-
range parts of the $G(r)$ functions, which suggests that the orthorhombic
strain is locally enhanced. In the neutron case, for the 2-7 Å range, the
$b$/$a$ ratio displays a plateau in the temperature range 75-105 K with the
extent of this plateau reaching up to 150 K when the $r$ interval is the
shortest. A similar tendency is also seen in the XRD data for the 2-7 Å range,
but in this case there are insufficient data points to determine the
temperature range of the plateau. Note that 75 K corresponds to the
temperature $T^{*}$, which is considered as the onset temperature of spin
fluctuations Ros2015 ; Ros2018 ; Gri2018 .

We should also note that in the 2-5 Å range, the $G(r)$ samples mainly
distances between atoms in the same FeSe plane while for larger $r$ ranges
inter-plane contacts have increasing weight in the $G(r)$. This result
suggests a larger crystallographic coherence of the orthorhombic distortion in
the $ab$ plane compared to the direction along the $c$ axis. Consistently, the
residuals are much smaller in the 2-5 Å fit than in the 2-7 Å range (e.g. at
90 K, $R_{w}$ is 0.029 and 0.059, respectively, for the neutron case). For the
$r$ ranges 2-7 Å and 2-5 Å a kink can be seen around 20 K, which coincides
with the electronic anomaly observed at $T^{**}$. Figs. 1 (b)-(e) display the
anisotropic displacement parameters U(Fe) and U(Se) refined in different $r$
ranges. To avoid correlations among the parameters, we assumed $U_{11}=U_{22}$
for both Fe and Se in the $G(r)$ refinements. The anisotropic displacement
parameters from the reciprocal space refinement are systematically higher than
those obtained from the refinement of the real-space. This difference is well
seen in both $U_{11}$ and $U_{33}$ and suggests that displacements of atoms
with respect to the average structure occur along and perpendicular to the
FeSe planes.

Figure 4: Neutron diffraction $G(r)$ peak positions (symbols) in the
temperature range 3-75 K compared with the corresponding distances obtained
from the reciprocal-space Rietveld refinement of the D2B data. The error bars
are smaller than the symbol size. Figure 5: (a)-(e) Lattice parameters at
different temperatures obtained from $G(r)$ as a function of $r$, using 10 Å
wide $r$ intervals. The $r$ values are the centroids of the range. Straight
lines are the lattice parameters obtained from the reciprocal-space Rietveld
analysis. Blue lines are used in panels (d) and (e) because $a\,=\,b$ in the
tetragonal phase. (f)-(j) Torsion angle of the FeSe4 tetrahedra as a function
of $r$ at different temperatures. The values obtained from the PDF (crosses)
replotted as a function of the $r$ range considered and compared to the
Rietveld results (black lines).

In the following, we describe the results of the real-space structural
analysis of synchrotron x-ray diffraction data in the temperature range 150–5
K. To visualize the local distortions described below, in Figs. 2(a) and (b),
we display schematic diagrams of FeSe unit cells in the tetragonal and
orthorhombic structures, while Figs. 2(c) and (d) show the corresponding
structures in the $ab$ plane. The figures depict the lattice parameters $a$,
$b$, $c$, the inter-atomic distances, and the torsion angle $\phi$ of the
FeSe4 tetrahedra. In Figs. 3(a) and (b), the short-range part of the $G(r)$
curves at different temperatures are presented. Due to the wavelength used and
the geometrical constraints of the cryostat ($2\theta_{max}=90^{\circ}$) only
the data up to $Q_{max}=25\,\mathrm{\AA}$ have been used. The peaks below
$r\,<\,3\,\mathrm{\AA}$ and the one around 4.5 Å have straightforward
attributions to single Fe–Se and Fe–Fe inter-atomic distances (see labels in
the figure). Conversely, the peaks in the 3.3–4.0 Å range contain three
different Se–Se contributions plus one from Fe–Fe. In all cases, peak
broadening is apparent upon lowering temperature. At 5 K, the peaks AFe-Se,
BFe-Fe and EFe-Se are broader than at 15 and 30 K, which is contrary to the
expected structural effect when the $G(r)$ peak width arises only from the
thermal atomic vibrations. The CFe-Se, C${}^{\prime}_{\mathrm{Se-Se}}$, and
DSe-Se peaks display some non-monotonic behavior, too. However, due to the
superposition of different contributions, it is more difficult to attribute
this finding to certain single inter-atomic distances.

The $G(r)$ peaks shown in Fig. 3 (a) and (b) have been first analyzed using
the so called direct analysis. Each inter-atomic distance has been fitted
using a Gaussian function allowing the area, position, and full width at half
maximum (FWHM) parameters to vary. The results are summarized in Figs. 3(c)
and (d). In Fig. 3(c), the peak positions are reported, while in Fig. 3(d),
the relative FWHM values are shown.

All the FWHM values increase upon cooling only for $T\leq\,15$ K. However, the
change in FWHM $\Delta=$[FWHM(5 K)- FWHM(15 K)] are less than 0.02 Å. The
nearest neighbor Fe-Fe distance (BFe-Fe) displays a comparatively larger value
of $\Delta\approx 0.018$ Å than the nearest neighbor Fe-Se (AFe-Se,
$\Delta\,\approx 0.016$ Å) and next-next-nearest neighbor Fe-Se (EFe-Se,
$\Delta\,\approx 0.007$ Å). Together with the atomic distances obtained from
the analysis of $G(r)$ (local structure), the corresponding distances acquired
from the reciprocal-space Rietveld analysis (average structure) are shown in
Fig. 3(c) for comparison. For all the distances considered, it can be noted:
(i) The shortest Fe–Se distance in the local structure is slightly shorter
than the corresponding distance in the average structure. (ii) The shorter
Fe–Fe distance in the local structure is always larger than that in the
average structure. (iii) The Fe–Se distance of 4.4 Å is the same in both local
and average structures. This trend is confirmed also by the real-space PDF and
reciprocal-space Rietveld analysis of the neutron diffraction data, see Fig.
4.

The very high angular resolution of the ID22 data allows calculating reliable
$G(r)$ functions up to tens of angström. Thus, we analyzed the experimental
$G(r)$ functions in the temperature range 5-150 K assuming the $Cmma$ lattice
structure at different $r$ intervals (box car refinements). Box car
refinements were carried out on $r$–intervals 10 Å wide. At all measured
temperatures, no marked deviations from the $Cmma$ structural model are
observed. An interesting result of the box car refinements is the trend of the
lattice parameters, as a function of $r$, as reported in Fig. 5 (a)-(e),
marked by symbols. In the same figure, reciprocal-space Rietveld results are
also given for comparison (dashed lines). The orthorhombic strain
$\propto\mid{a-b}\mid$ seems to be larger than the average strain at low $r$
values and it reaches the latter value within 20-30 Å. We would like to note
that, for $T\,>\,90$ K, i.e., above $T_{s}$, large in-plane orthorhombic
strain coexists with weak inter-plane one. For this reason, in the refinements
at the lowest $r$ values and $T\,\geq\,90$ K shown in Fig. 5, two not matching
orders coexist, which leads to less stable refinements.

The orthorhombic strain is related to the torsion angle of the FeSe4
tetrahedra, $\phi=2\mathrm{tan}^{-1}(a/b)$, as defined in Ref. [Mc2009a, ]. It
is a coherent twisting of the upper and lower Se pairs that make up the Fe-Se
tetrahedron; the smaller value of the angle $\phi$ corresponds to the larger
orthorhombic strain. In Fig. 5(f)-(j), the torsion angles $\phi$ are plotted
as a function of $r$. For almost all temperatures up to 150 K, at low $r$, the
$\phi$ values are smaller than those in the average structure shown by the
straight lines. This confirms our findings from the real-space analysis of the
neutron data. However, within a few nm, the $\phi$ values merge into the
average angle at all temperatures, and above 90 K the average structure
appears to be tetragonal.

Remarkably, although the nematic phase transition occurs in FeSe at
$T_{s}\,\approx\,$90 K, at which the average crystal structure changes from
tetragonal to orthorhombic phase, recent PDF studies of FeSeKoch2019 ;
Benj2019  have shown that the short-range orthorhombic distortion exists
already at 300 K and extends over a length scale of 10-30 Å. We considered
both $Cmma$ and $P4/nmm$ models for the full temperature range studied here.
For the neutron diffraction data taken at 298 K, the $Cmma$ model provided
lower residual values than the $P4/nmm$ for $r$ values as small as 5 Å. Our
results are thus in full agreement with the earlier PDF work in Ref.
[Koch2019, ].

Finally, with the intention to investigate whether there is any deviation from
$Cmma$ symmetry at low temperatures, we considered additional models for the
PDF analysis. Based on their electron diffraction studies on Fe1.01Se McQueen
$et\,al.$,Mc2009a  proposed that the crystal structure at 11 K is likely of
lower symmetry than that of the $Cmma$ space group. They put forward two
distortion modes of the in-plane Fe lattice. Either there are alternating
displacements of Fe in direction of the short $a$-axis, which may be driven by
metal-metal dimer formation, or there are displacements of Fe nearest
neighbour-pairs along the long $b$-axis direction. In the latter case, the Fe
move alternatingly perpendicular to the distance vector of the pairs that is
oriented along the short $a$-direction. This mode avoids dimer formation, but
rotates the pairs while keeping their distance. For both modes, the
distortions preserve the C-centering, but remove the glide plane. Combinations
or partial (staggered) activation of these two distortion modes would display
dimerized and rotated Fe-pairs. Such more complex lattice distortions
simultaneously break both the glide mirror and the C-centering Mc2009a . These
distortions are subtle and were not observed in the synchrotron x-ray
diffraction reported in Ref. [Mc2009a, ]. Our PDF analysis shows that these
in-plane distortions vanish in the $r$ range of 2–7 Å and both the models do
not provide any improvement in the quality of the fit. We also verified the
effect on the refinement of some other possible distortions such as three
different types of buckling in which, (i) the Fe $z$ coordinate is shifted up
and down forming a checkerboard, (ii) the Fe $z$ coordinate is shifted up and
down forming stripes running along $y$, and (iii) the Fe $z$ coordinate is
shifted up and down forming stripes running along $x$. Also, a structural
model with a $1\times 1\times 2$ supercell formation was tried. However, the
quality of the corresponding fits did not support these scenarios.

For the purpose of correlating the electronic properties of FeSe with the
structural details determined in this study, we plot in Fig. 6 the resistivity
$\rho(T)$ and the Hall coefficient $R_{H}(T)$, taken from the Refs. [Lin2016,
] and [Ros2015, ]. As summarized in the figure, above the structural
transition temperature $T_{s}$, the average structure is tetragonal, but local
incoherent orthorhombic distortions were observed. However, below $T_{s}$,
although electronic anomalies are visible in Fig. 6 at $T^{*}$ and $T^{**}$,
reduction in symmetry below $Cmma$ was not found even at shorter length-
scales. Nonetheless, on a local scale, some strain in the lattice was observed
as small anomalies in the temperature dependence of the $b/a$ ratio and inter-
atomic distances. The average structure remains orthorhombic down to the
lowest measured temperature of 3 K.

##  IV Discussion

Our structural study involving both reciprocal-space and real-space analysis
of the scattering data indicate that the temperature evolution of the static
FeSe structure from room temperature through the nematic phase transition at
$T_{s}$ and through further anomalies at lower temperatures is smooth. Within
the experimental resolution, the structural phase transition from the
tetragonal to the orthorhombic symmetry is the only transformation in the
system. It leads to a macroscopic symmetry breaking well above the transition
to the superconducting state. This transition has peculiar features, as seen
from the fact that orthorhombic distortions are present in the form of
nanodomains already above $T_{s}$. Further, as discerned by the $b/a$ ratio in
Fig. 1, on a short length scale (especially at $r=2-5$ Å that mainly samples
the intraplane interatomic distances), the orthorhombic deviation from
tetragonal symmetry is always much stronger than the average value. This
behavior suggests that the orthorhombic ordering is highly anisotropic, having
a larger coherence length within each FeSe layer than along the $c$ direction.

Figure 6: Resistivity $\rho(T)$ and the Hall coefficient $R_{H}(T)$ of a FeSe
single crystal replotted from the Refs. [Lin2016, ] and [Ros2015, ],
respectively. The temperatures $T_{s}$, $T^{*}$, $T^{**}$, and $T_{c}$
represent the temperatures of the structural phase transition, onset of spin-
fluctuations, onset of the pseudogap, and the superconducting transition
temperature, respectively. In our studies, we observe a plateau in the $b/a$
ratio at $T^{*}$, and some broadening in the peaks corresponding to Fe–Fe,
Fe–Se, and the next-nearest Fe–Se bond distances at $T^{**}$, but the average
structure remains orthorhombic below $T_{s}$.

Above $T_{s}$, the orthorhombic domains show coherence lengths of a few nm
within the $ab$-planes, but almost no coherence along the $c$-axis. From the
perspective of chemical bonds and the electronic structure of FeSe, such a
behavior is plausible, as the microscopic origin for the nematic ordering
likely pertains to electronic correlations within individual FeSe-layers. A
weak coherence along the $c$-direction is also plausible, as the cohesion of
the FeSe-layers has a different character with prominent contributions of the
van-der-Waals bonding Ricci2013 ; Loch2021 . Therefore, the ordering into a
three-dimensional long-range order may evolve first through an electronic
instability within the FeSe-planes that is already present at high
temperaturesSko2018 . The concomitant square-to-rectangle strains in the
$ab$-planes of the tetragonal lattice cannot grow to macroscopic length due to
the lack of coherence between the FeSe-planes. Such a mixture of local
rectangular lattice patches in the absence of coherence in $c$-direction will
appear tetragonal in the average crystal structure. The transition to the
long-range ordering at 90 K then requires an anisotropic divergent growth of
domains in the $ab$ plane and possibly a different growth rate of coherence
with decreasing temperature in the $c$-direction, as indicated in a recent NMR
experimentWei2021 .

Our analysis of the difference between the local orthorhombic distortion
measured in real space and the average lattice parameters $a$ and $b$ obtained
in the reciprocal space support such a picture of an anisotropic
transformation into nematic long-range order. However, there is an additional
feature. The data in Fig. 5 display a tendency to local orthorhombic
distortion $b/a$ larger than the average on short lengths below 5 Å even below
$T_{s}$. This may indicate that the microscopic nematic instability is
frustrated by some additional mechanism and that the nematic state retains a
partially incomplete or incoherent character at low temperatures. Further, as
there are no important modifications of the lattice structure discernible at
and below the transition to the superconducting state at 8 K, effects of
superconducting ordering on the lattice or modifications of the nematic state
are not observed at least down to 3 K.

##  V Conclusions

We presented a comprehensive structural analysis of FeSe in the temperature
range of 298–3 K using synchrotron x-ray and neutron diffraction measurements.
Although the global reduction in symmetry from $P4/nmm$ to $Cmma$ was observed
only below 90 K, short-range orthorhombic correlations are found already at
room temperature, which is consistent with previously published results
Koch2019 ; Benj2019 ; Kon2019 ; Wang2020 . No changes in the short-range
correlations were observed when passing through the nematic phase transition
from high temperature. A plateau in the $b/a$ ratio was observed at short-
length scales around $T^{*}$. At the onset temperature of a putative pseudogap
$T^{**}$, some broadening in the peaks corresponding to Fe–Fe, Fe–Se, and the
next-nearest Fe–Se bond distances, and tetrahedral torsion angles were found
in the PDF analysis of x-ray diffraction. Our studies show that the local
symmetry remains $Cmma$ also in the superconducting state down to at least 3
K. The local orthorhombic symmetry spanning three orders of magnitude in
temperature is yet another trait of FeSe.

###### Acknowledgements.

We thank ESRF and ILL for granting beam time. We are grateful to Catherine
Dejoie and Andy Fitch for providing excellent technical support during our
measurements at ID22, ESRF. We thank U. Nitzsche for technical assistance.

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