# Structural Evolution and Onset of the Density Wave Transition in the CDW
Superconductor LaPt2Si2 Clarified with Synchrotron XRD

E. Nocerino  KTH Royal Institute of Technology, Department of Applied Physics,
Alba Nova University Center, Stockholm, SE-114 21, Sweden<EMAIL_ADDRESS>I.
Sanlorenzo  Nanoscience Center, Niels Bohr Institute, University of
Copenhagen, Nørre Allé 59, 2100 Copenhagen, Denmark  K. Papadopulos
Department of Physics, Chalmers University of Technology, SE-412 96 Göteborg,
Sweden  M. Medarde  Laboratory for Multiscale Materials Experiments, Paul
Scherrer Institute, CH-5232 Villigen PSI, Switzerland  J. Lyu  Laboratory for
Multiscale Materials Experiments, Paul Scherrer Institute, CH-5232 Villigen
PSI, Switzerland  Y. M. Klein  Laboratory for Multiscale Materials
Experiments, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland  A.
Minelli  Inorganic Chemistry Laboratory, University of Oxford, Oxford OX1 3QR,
United Kingdom  Z. Hossain  Department of Physics, Indian Institute of
Technology, Kanpur 208016, India  A. Thamizhavel  DCMPMS, Tata Institute of
Fundamental Research, Mumbai 400005, India  K. Lefmann  Nanoscience Center,
Niels Bohr Institute, University of Copenhagen, Nørre Allé 59, 2100
Copenhagen, Denmark  O. Ivashko  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  M. v. Zimmermann  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Y. Sassa  Department
of Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden  M.
Månsson  KTH Royal Institute of Technology, Department of Applied Physics,
Alba Nova University Center, Stockholm, SE-114 21, Sweden

###### Abstract

The quasi-2D Pt-based rare earth intermetallic material LaPt2Si2 has attracted
attention as it exhibits strong interplay between charge density wave (CDW)
and and superconductivity (SC). However, the most of the results reported on
this material come from theoretical calculations, preliminary bulk
investigations and powder samples, which makes it difficult to uniquely
determine the temperature evolution of its crystal structure and,
consequently, of its CDW transition. Therefore, the published literature
around LaPt2Si2 is often controversial. In this paper, we clarify the complex
evolution of the crystal structure, and the temperature dependence of the
development of density wave transitions, in good quality LaPt2Si2 single
crystals, with high resolution synchrotron X-ray diffraction data. According
to our findings, on cooling from room temperature LaPt2Si2 undergoes a series
of subtle structural transitions which can be summarised as follows: second
order commensurate tetragonal ($P4/nmm$)-to-incommensurate structure followed
by a first order incommensurate-to-commensurate orthorhombic ($Pmmn$)
transition and then a first order commensurate orthorhombic ($Pmmn$)-to-
commensurate tetragonal ($P4/nmm$). The structural transitions are accompanied
by both incommensurate and commensurate superstructural distortions of the
lattice. The observed behavior is compatible with discommensuration of the CDW
in this material.

## Introduction

A grand challenge in condensed matter physics is understanding the mechanisms
underlying high temperature superconductivity (SC). Materials with competing
electron spectrum instabilities, such as Cooper pairing and charge/spin-
density waves (CDW/SDW), represent the ideal playground for this kind of
investigations since the electron-phonon coupling established in such systems
is believed to be a key factor in inducing SC. The quasi-2D Pt-based rare
earth intermetallic material LaPt2Si2 belongs to this family of compounds as
it exhibits strong interplay between CDW and SC. LaPt2Si2 crystallizes in a
CaBe2Ge2-type tetragonal structure (space group $P4/nmm$), where two
nonequivalent layers (Si1–Pt2–Si1) and (Pt1–Si2–Pt1) are arranged in
alternating stacking separated by lanthanum atoms (Fig. 1). In single-phased
powder samples, indications of a first-order transition were observed from
high temperature tetragonal to low temperature orthorhombic phase, accompanied
by a CDW transition around TCDW = 112 K, where superlattice reflections
corresponding to (n/3, 0, 0), with n = 1 and 2 were observed, followed by a SC
transition at Tc = 1.22 K [1, 2]. It was suggested that the CDW modulation of
the crystal lattice would induce a tripling of the initial unit cell and that
it would propagate in the P2 layer while superconductivity would be
established in the P1 layer [2]. The Fermi surface of LaPt2Si2 was found to
have a two-dimensional nature [3] and theoretical calculations of the phonon
dispersion predict phonon-softening instabilities, leading to structural
instabilities that are compatible with the (1/3, 0, 0) $q$-vector of the CDW.
However, it was also shown that CDW and SC should coexist in the Pt1 layer [3]
and the softened phonon modes would mainly arise from Pt1. This finding
suggested that the CDW transition occurs in the Pt1 layers, with large
electron-phonon interaction [4]. The latter suggestion was later confirmed by
Pt-NMR measurements [5], contrary to the aforementioned conjecture of Nagano
[2].

Figure 1: Room temperature crystal structure of LaPt2Si2. The alternating Pt-
Si planes are explicitly labelled.

Beyond the results on polycrystalline samples [6], single crystal diffraction
studies with in-house characterization methods were also reported. According
to these studies the CDW transition occurs at 85 K (slightly lower than the
112 K reported from studies on polycrystalline samples) in correspondence to
the maximum intensity of the superlattice satellites, which were found to have
propagation vector (0.36 0 0), slightly different from the reported (1/3 0 0),
indicating that the CDW modulation is incommensurate. Also, there was no clear
indication of a structural transition towards orthorhombic crystal symmetry in
single crystalline LaPt2Si2, showing that the properties of this material as a
single crystal are slightly different from the ones as a polycrystal [7].
Additionally, the CDW-induced modulation of the lattice has recently been
found to develop a periodicity also along the $c$-axis with a different
propagation vector and a different temperature dependence with respect to the
previously identified one, propagating in the $ab$-plane [8] which seems to
indicate that multiple CDWs develop in this material. Finally, the
superconductivity in this material, putatively non-conventional, (the Fermi
surface exhibits 2 gaps of different magnitude according to $\mu^{+}$SR
measurements [9]) was recently found to have SC gap and London penetration
depth well described by standard BCS models [10].

The published literature about the properties of this sample is slightly
inconsistent and sometimes contradictory. At present, no experiments have
measured the phonon dispersion and confirmed the theoretical predictions, nor
provided direct evidence of the CDW transition temperature dependence, while
the low temperature crystal structure in the CDW phase is not solved for
powder nor for single crystalline LaPt2Si2. In this paper we present the
results of a synchrotron XRD investigation on LaPt2Si2 single crystals, and
clarify the complex evolution of the crystal structure in LaPt2Si2 as well as
the temperature dependence and nature of its CDW state.

## Results

###  0.1  Synchrotron XRD measurements

Due to over-saturation of the structural Bragg peaks in the atten0 setting,
the atten2 data were used for structural analysis and determination of the
temperature dependence of the satellites. The atten0 data are presented in
these paper for clarity of display of weak satellites and diffuse scattering.
Figure 2 a) shows the $h$ $k$ $0$ (atten2), $h$ $k$ $0.5$ (atten0) and $h$ $k$
$0.5$ (atten0) planes in the reciprocal space at 300 K. Well structured
diffuse scattering, indicating the presence of interactions with short-range
correlation lengths induced by charge ordering, is already present at T = 300
K. Since the diffuse scattering does not converge on Bragg peaks at integer
$h$ $k$ $l$, but it rather lies around them at semi-integer positions, the
possibility of thermal diffuse scattering is excluded.

This implies that the charge ordering that results in the multi-$q$ modulation
of the lattice is already in place in this temperature range. Therefore, the
onset of the charge ordering in LaPt2Si2 is well above room temperature. The
atten0 setting was chosen to display the $h$ $k$ $0.5$ and $h$ $k$ $0.25$
planes to highlight the presence of the diffuse scattering, since no
structural Bragg peaks are allowed in this region of the reciprocal space and
their over-saturation would not cause too much disturbance. As the temperature
decreases, we observe the appearance of satellites with 6 different
propagation vectors: between 235 K and 35 K the satellites have average values
of the propagation vectors $q\prime 1$ = [0.35736 0 0] $\approx$ [0.36 0 0]
and $q\prime\prime 1$ = [0 0.35905 0] $\approx$ [0 0.36 0] in the full
temperature range. Below 35 K the $q1-q\textquoteright 1$ satellites become
weaker (but not enough to completely disappear) and the formation of the
satellites $q\prime 2$ = [0.18 0.18 0.5] and $q\prime\prime 2$ = [0.18 -0.18
0.5] occurs. Looking at the reciprocal unit cell of the atten0 data at base
temperature a third set of peaks could be identified with propagation vectors
$q\prime 3$ = [0.3 0.3 0.25] and $q\prime\prime 3$ = [-0.3 -0.3 0.25]. This
new set of satellites, whose intensity might possibly become stronger at lower
temperature, could be noticed only by exploring the reciprocal cell in the
zero attenuation data. Once they have been identified, appropriate cuts in the
atten2 data allowed the observation of the $q3$ satellites in this dataset as
well, despite the stronger attenuation conditions (Fig. 2 b)).

Figure 2: Reciprocal space planes [$h$ $k$ $0$], [$h$ $k$ $0.5$], [$h$ $k$
$0.25$] and [$h$ $h$ $l$] at a) 300 K, showing the room temperature short-
range diffuse scattering, and b) 92 K and 24 K, showing the 3 sets of
satellites $q1$ (marked with the red color), $q2$ (marked in blue) and $q3$
(marked in green) as they appear. The first three cuts are therefore done by
keeping the $hl$-plane constant, while shifting the $l$-axis in half-integer
steps. The fourth cut is done along the $l$-axis in the $hk$ diagonal
direction, to allow the simultaneous visualization of the three orders of
satellites at 24 K. The origin of the different plots is marked at the bottom
for each of them.

The reciprocal space planes $h$ $k$ $l$ are obtained by reconstructing the
diffraction data in a single layer defined by the L1 = (1 0 0) and L2 = (0 1
0) vectors with the origin in (0 0 0), (0 0 0.5) and (0 0 0.25) respectively,
while the reciprocal space planes $hhl$ are obtained by reconstructing the
diffraction data in a single layer defined by the L1 = (1 1 0) and L2 = (0 0
1) vectors with the origin in (2.18 -2.18 0.5), resulting in a plane parallel
to the $l$ direction oriented along the $k=h-4.36$ direction in the $hk$
plane, and with the origin in (0.7 -3.7 0.25), resulting in a plane parallel
to the $l$ direction oriented along the $k=h-3$ direction in the $hk$-plane.
The latter choice for the reciprocal lattice cut allows clear simultaneous
observation of satellites belonging to all the 3 sets of propagation vectors.
All the satellites show sharp line profiles along the 3 directions $h$, $k$
and $l$, which indicates that such lattice modulations have a long
3-dimensional phase coherence length for all the propagation vectors, which
would imply strong coupling of the Pt layers. It should be noted that, while
the appearance of the $q1$ and $q2$ satellites is preceded by diffuse
scattering, no detectable diffuse scattering is observed in correspondence of
the $q3$ satellites sites prior to their formation. By following the
temperature evolution of the intensities and full width at half maximum of the
satellites [$q\textquoteright 1$ \+ (0 -4 0)], [$q\textquoteright 2$ \+ (2 -2
0)] and [$q\textquoteright 3$ \+ (-2 3 0)] (Fig. 3), four main temperature
regimes have been identified: between 300 K and T1 = 230 K a well structured
diffuse scattering corresponding to the $q1$ and $q2$ satellites positions is
observed, between T1 = 230 K and T2 = 110 K the $q1$ satellites are formed and
their intensity increases as the temperature decreases until 110 K, where an
abrupt drop in the satellites intensity is observed. Between T2 = 110 K and T3
= 60 K the $q2$ satellites are formed, their intensity grows as the
temperature decreases and they coexist with the $q1$ satellites until 60 K,
where an abrupt drop in both the $q1$ and $q2$ satellites intensity is
observed. Below T3 = 60 K the $q3$ satellites are formed, their intensity
grows as the temperature decreases and they coexist with the $q2$ satellites,
whose intensity shows a similar trend as the $q3$ satellites. The values for
T1 and T2 are chosen as the onset of the falling edge of the FWHM temperature
dependence for the $q1$ and $q2$ satellites, i.e. when the peaks become sharp.
All these periodic distortions of the lattice seem to have a different nature,
which suggests that the origins of their formation are different. The $q1$
satellites are characterized by incommensurate propagation vectors with zero
component along the $c$-axis, they are connected to the corresponding nesting
wave-vectors $Q$ = (1/3 0 0) which, according to previously reported
theoretical calculations [4], modulate the Fermi surface in LaPt2Si2 by
creating gaps at these positions resulting in the CDW state. The $q2$
satellites are characterized by propagation vectors with incommensurate
components in the $ab$-plane and a commensurate component along the $c$-axis.
The fact that the $q2$ propagation vectors can be expressed as linear
combinations of the $q1$ vectors as:

$\displaystyle q\prime 2$ $\displaystyle=\frac{q\prime 1+q\prime\prime
1+(001)}{2},$ (1) $\displaystyle q\prime\prime 2$ $\displaystyle=\frac{q\prime
1-q\prime\prime 1+(001)}{2}$ (2)

seems to indicates that the $q2$ satellites are not due to an independent CDW
$Q$ vector, but are rather related to the $q1$ satellites. The appearance of
the $q2$ satellites is also accompanied by the formation of an out-of-plane
commensurate modulation of the cell, probably needed to compensate the
structural instability induced by the $q1$ modulation. The $q3$ satellites are
characterized by propagation vectors which are linearly independent from $q1$
and $q2$, but their coordinates in the reciprocal space can be expressed in
terms of rational fractions of the reciprocal lattice vectors as:

$\displaystyle q\prime 3$
$\displaystyle=(\frac{10}{3}(110)+\frac{1}{4}(001)),$ (3) $\displaystyle
q\prime\prime 3$ $\displaystyle=(-\frac{10}{3}(110)+\frac{1}{4}(001))$ (4)

resulting in a superstructural cell which is quadrupled along the $c$-axis.
The $q3$ super-cell seems to be just a consequence of the structural
instabilities introduced by the incommensurate modulations of the unit cell,
and it does not seem to be related to any additional CDW nesting vector.
However, the very weak intensity of the $q3$ satellites, makes the refinement
of the $q3$ propagation vectors very difficult. Diffraction measurements at
temperatures lower than 24 K, where the intensity of the $q3$ satellites is
expected to be enhanced, would be needed for precise determination of their
position.

The CDW in this material was previously reported to show indications of a 2
dimensional nature and proposed to develop in the P1 layer only. However, our
observation of 3-dimensional super-lattice distortions and the long range
correlation between the Pt layers seems to point towards a more uniform
propagation of the CDW in LaPt2Si2 which probably develops also in the P2
layer.

Figure 3: Temperature dependence of the integrated intensities (left y axis)
and FWHM (right y axis) for 3 satellites belonging to the 3 modulations of the
lattice $q1$, $q2$ and $q3$, as for the labels on each panel. The transition
temperatures T1, T2 and T3 are explicitly marked in magenta.

Regarding the evolution of the crystal structure, a change in the lattice
parameters occurs as a function of temperature so that the $b$-axis becomes
gradually longer than the $a$-axis. Figure 4 displays the temperature
evolution of the difference $b-a$. Just like for the satellites intensities,
also in this case four temperature regimes can be identified: between 300 K
and T1 = 230 K $b-a$ is close to zero within the error bars and the crystal
structure of LaPt2Si2 can be reliably refined with the tetragonal space group
$P4/nmm$ (no. 129), between T1 = 230 K and T2 = 110 K the unit cell is
gradually distorted on cooling as the difference $b-a$ grows. In this
temperature range the $q1$ lattice modulation is incompatible with the
tetragonal symmetry and induces a strain resulting in the $a$ and $b$ axes
being inequivalent. Refinement attempts with space group $P4/nmm$ could not be
stabilized. Therefore, from a group subgroup relationship argument, the
centrosymmetric orthorhombic space group $Pmmn$ (no. 59) was identified as the
most probable for a structural transition from $P4/nmm$, however, refinement
attempts with this and other orthorhombic space groups (namely the non
centrosymmetric $Pmn2$ and $Cmme$) could not provide a stable result either.
Indeed, calculations for crystal system recognition including the main unit
cell and the $q1$ modulation vectors for the T = 140 K data, performed with
the software Jana, provided the triclinic and monoclinic crystal systems as
the only possible symmetry choices which would make the main unit cell
consintent with the $q1$ modulation. However, even though in principle the
proposed triclinic and monoclinic ($P2_{1}/m$, $P2/m$, $P2_{1}$, $P2$) space
groups are maximal subgroups of the tetragonal $P4/nmm$, in practice the path
of symmetry reduction from the parent structure to the lower symmetry space
groups requires several intermediate groups. This makes the physical
realization of the structural transition highly improbable, according to the
Symmetry Principle. The most reasonable among the monoclinic subgroups is
$P2_{1}/m$, which can be obtained from the aristotype through the path
$P4/nmm\rightarrow Pmmn\rightarrow P2_{1}/m$. However, attempts of refining
the T = 140 K data with the latter monoclinic space group did not provide any
stable result either. Therefore, the structural evolution in the temperature
range between T1 = 230 K and T2 = 110 K will be described as a second order
transition from a commensurate tetragonal phase to an incommensurate distorted
phase in which the $b$-axis is elongated with respect to the parent tetragonal
structure. The expression "distorted tetragonal" will be used from now on to
identify such a phase. Around a temperature T2 = 110 K a first order
transition occurs between the incommensurate distorted tetragonal phase and
the centrosymmetric orthorhombic $Pmmn$ space group, along with a slight
increase in the unit cell volume. The refinement of the crystal structure in
this phase could be reliably stabilized with the best R factor achieved for
the T = 102 K data set (see table 1). The R factor becomes progressively worse
on cooling from T = 102 K to T = 64 K. Such a structural transition is not of
displacive nature, but more of a order-disorder type. Indeed, the small value
of the total amplitude of the displacive distortion ($A$ = 0.0072 Å) leaves
the atomic positions of the parent structure almost unchanged in the
orthorhombic cell, while the occupation probabilities of the atomic sites are
different (see table 1). It should be noted that for the refinement of the
commensurate main phase the satellite Bragg reflections are not taken into
account, therefore the low temperature crystal structures reported here are
average crystal structures. Between T2 = 110 K and T3 = 60 K the orthorhombic
symmetry is maintained, in correspondence to the coexistence of $q1$ and $q2$
satellites. Below T3 = 60 K a first order transition occurs and the tetragonal
symmetry $P4/nmm$ is restored. Therefore, the structural evolution of LaPt2Si2
can be summarized as follows on cooling: tetragonal $P4/nmm$ $\Rightarrow$
distorted tetragonal (2nd order) $\Rightarrow$ orthorhombic $Pmmn$ (1st order)
$\Rightarrow$ tetragonal $P4/nmm$ (1st order).

Figure 4: Temperature dependence of the difference between the $b$-axis and
$a$-axis in the full temperature range. The evolution of the lattice
distortions is clearly visible and the transition temperatures T1, T2 and T3
are explicitly marked.

Table 1 reports the results of the refinement within the 3 temperature regimes
[T $<$ T3], [T3 $<$ T $<$ T2] and [T $>$ T1], along with their respective
reliability factors R1. The values of the R-factor for the 3 refinements,
below 5$\%$, indicate a good agreement between the calculated and observed
models. The refinement showed significant improvement with the adoption of an
anisotropic displacement parameter U. This implies a propensity for the atoms
in LaPt2Si2 to move away from their reference lattice positions along certain
privileged directions. In particular, from the values of the diagonal elements
of the tensor U reported in table 1 (the refinement provided zero value for
the off diagonal terms), the atoms Pt1 dislocate preferentially along the
$c$-axis while the Pt2 within the $ab$-plane. A graphic representation of the
anisotropic displacement ellipsoids extracted from the 102 K refinement
results is displayed if figure 5. This behavior seems to indicate that the Pt2
atoms are responsible for the in-plane coupling of the CDW propagation while
the Pt1 atoms are responsible for the out of plane coupling along the
$c$-axis.

Figure 5: Unit cell of LaPt2Si2 with the displacement ellipsoids of the
respective atoms. The probability for each atom to be included in the
ellipsoids was set to 100 $\%$

This fact is in contrast with the theoretical calculations and experimental
evidences according to which the Pt1 layer is the only responsible for the
occurrence of the CDW state in LaPt2Si2 [4, 5]. Indeed our results seem to
indicate that 2 CDW transitions occur, one at T = T1 = 230 K with nesting
vectors $q1$, propagating in the Pt2 layer (since $q1$ has $l$ component equal
to zero), and a second one at T = T2 = 110 K with nesting vectors $q2$,
propagating in the Pt1 layer (since $q1$ has non-zero $l$ component). The
reason for this discrepancy might be that the NMR data, which provide proof of
the fact that the Pt1-5d bands are the only responsible for the CDW transition
in LaPt2Si2 [5], were only acquired in a temperature range from 5 K to 200 K,
therefore they observed the T2 transition, but not the T1. An NMR experiment
in a wider temperature range might confirm or disprove this conjecture. It
should be noted that a system analogous to LaPt2Si2 with a CaBe2Ge2-type
structure, SrPt2As2, shows a double CDW transition occurring in two separate
layers [11].

Table 1:  Structural parameters in LaPt2Si2 Summary of the space groups, unit
cell parameteres, atomic coordinates and sites, anisotropic Debye waller
factors and reliability $R$-factors of the refinement for LaPt2Si2 in the
three temperature regimes T $>$ T1, T3 $<$ T $<$ T2 and T $<$ T3.

T < T3 | T = 24 K | SG: $P4/nmm$ |  | a = b = 4.2727 (Å), | c = 9.7987 (Å), |  $\alpha=\beta=\gamma$ = 90∘ |  |  |
---|---|---|---|---|---|---|---|---|---
| x | y | z | Occ | site, symm | U11 | U22 | U33 |
Pt1 | 0.75000 | 0.75000 | 0.62009(4) | 1 | 2c, 4mm | 0.00150(8) | 0.00150(8) | 0.00815(8) |
Pt2 | 0.75000 | 0.25000 | 1.00000 | 1 | 2a, -4m2 | 0.0189(2) | 0.0189(2) | 0.0045(2) |
La | -0.25000 | 0.75000 | 0.25613(6) | 1 | 2c, 4mm | 0.0022(1) | 0.0022(1) | 0.0069(2) |
Si1 | 0.75000 | 0.75000 | 0.8707(5) | 1 | 2c, 4mm | 0.0113(11) | 0.0113(11) | 0.010(2) |
Si2 | 0.25000 | 0.75000 | 0.50000 | 1 | 2b, -4m2 | 0.0027(7) | 0.0027(7) | 0.0092(14) |
R1 ($\%$) |  |  |  | 4.9 |  |  |  |  |
T3 < T < T2 | T = 102 K | SG: $Pmmn$ |  | a = 4.2981 (Å), | b = 4.3021 (Å), | c = 9.876 (Å), |  $\alpha=\beta=\gamma$ = 90∘ |  |
| x | y | z | Occ | site, symm | U11 | U22 | U33 |
Pt1 | 0.75000 | 0.75000 | 0.61995(4) | 0.5 | 2a, mm2 | 0.0052(2) | 0.0026(3) | 0.0073(3) |
Pt2 | 0.75000 | 0.25000 | 1.00011(3) | 0.5 | 2b, mm2 | 0.01195(20) | 0.0204(3) | 0.0062(3) |
La | -0.25000 | 0.75000 | 0.25612(7) | 0.5 | 2a, mm2 | 0.0057(2) | 0.0028(3) | 0.0071(3) |
Si1 | 0.75000 | 0.75000 | 0.8703(5) | 0.5 | 2a, mm2 | 0.010(2) | 0.0084(22) | 0.011(2) |
Si2 | 0.25000 | 0.75000 | 0.5001(2) | 0.5 | 2b, mm2 | 0.0027(9) | 0.0068(15) | 0.0088(15) |
R1 ($\%$) |  |  |  | 4.19 |  |  |  |  |
T > T1 | T = 300 K | SG: $P4/nmm$ |  | a = b = 4.2793 (Å), | c = 9.8119 (Å), |  $\alpha=\beta=\gamma$ = 90∘ |  |  |
| x | y | z | Occ | site, symm | U11 | U22 | U33 |
Pt1 | 0.75000 | 0.75000 | 0.61990(3) | 1 | 2c, 4mm | 0.0044(1) | 0.0044(1) | 0.0088(2) |
Pt2 | 0.75000 | 0.25000 | 1.00000 | 1 | 2a, -4m2 | 0.0144(1) | 0.0144(1) | 0.0086(2) |
La | -0.25000 | 0.75000 | 0.25568(5) | 1 | 2c, 4mm | 0.0049(1) | 0.0049(1) | 0.0085(2) |
Si1 | 0.75000 | 0.75000 | 0.8707(3) | 1 | 2c, 4mm | 0.0090(8) | 0.0090(8) | 0.011(1) |
Si2 | 0.25000 | 0.75000 | 0.50000 | 1 | 2b, -4m2 | 0.0067(5) | 0.0067(5) | 0.008(1) |
R1 ($\%$) |  |  |  | 4.3 |  |  |  |  |

The structural instability associated to the formation of CDW eventually
results in a structural transition from distorted tetragonal to orthorhombic
symmetry in LaPt2Si2, where ion displacements are needed to stabilize the
charge perturbation. Ionic relocation requires strong electron-phonon
coupling, which implies that the Fermi surface nesting/gapping cannot be the
only driving mechanism involved in this transition, denoting the fact that the
CDW established in LaPt2Si2 cannot be described within the simple Peierls
picture. A normal-to-incommensurate structural transition is considered to be
strong evidence of CDW formation and it was indeed observed in the
prototypical 2 dimensional layered CDW compound 2H-TaSe2 [12] as well as in
other systems that exhibit 2 dimensional and 3 dimensional CDW [13] [14]. The
phenomenology in these cases foresees an alteration of the normal crystalline
periodicity of the material caused by the CDW wave vector which, being
determined by the Fermi surface nesting, is not necessarily an integral
fraction of the reciprocal lattice vector. In this way an incommensurate
structure is established in the crystal, as a result of the distortion of the
parent commensurate structure, until the lattice undergoes a second distortion
towards a "locked-in" commensurate structure, whose reciprocal lattice vector
is an integral multiple of the CDW wave vector. Such structural changes are
expected to be second order (normal-to-incommensurate) and first order
(incommensurate-to-commensurate) respectively, according to the Landau theory
of phase transitions, as demonstrated by McMillan in 1975 [15]. In LaPt2Si2
however, although we observe the occurrence of the second order normal-to-
incommensurate and first order incommensurate-to-commensurate structural
transitions (tetragonal $P4/nmm$ \- distorted tetragonal - orthorhombic
$Pmmn$), the lowest energy state with a stable structure does not seem to be
achieved in this system within the investigated temperature range. Indeed, an
additional first order structural transition occurs below T = 60 K. By
refining the unit cell parameters together with the incommensurate $q1$
vectors across the temperature range where the $q1$ satellites are clearly
visible (70 K $<$ T $<$ 205 K), it is possible to plot the degree of
incommensurability $\delta$, defined as $q\prime 1$ = ($\frac{1}{3}+\delta$ 0
0), for the position of the $q\prime 1$ satellites relative to the main
lattice as a function of temperature (Fig. 6).

Figure 6: Incommensurability of the $q\prime 1$ satellites with respect to the
main lattice expressed in reciprocal lattice units. The solid line is a guide
to the eye.

The value of $\delta(T)$ reaches a minimum between 110 K and 85 K, which
corresponds to the temperature range in which the structural refinement with
the orthorhombic unit cell for the main phase shows the best reliability
factors (highlighted by the shaded region in figure 6), but it is never equal
to zero. Indeed the $\delta(T)$ value below 85 K increases again, in
correspondence to the increase in intensity of the higher order satellites
$q2$ (see figure 3). Although a structural transition to a crystalline
commensurate phase can be stabilized in the temperature range 85 K $<$ T $<$
110 K, it cannot be labelled as a "locked-in" transition because the CDW wave
vectors are never commensurate to the main phase, and indeed an additional
structural transition occurs at lower temperature. To interpret such a
behavior the concept of discommensuration (DC) might be relevant in this case.
DC was initially introduced theoretically by McMillan to explain the
properties of 2H-TaSe2 [16]; here the incommensurate phase is regarded as a
defect melting transition in which narrow domain walls separate large
commensurate domains and, within such narrow domains, the superlattice phase
fluctuates rapidly. Later on this conjecture was confirmed with a dark-field
electron microscopy experiments [17], which provided direct observation of the
commensurate and incommensurate domains and demonstrated that the CDW
dislocation is the main responsible of the normal-to-incommensurate transition
in 2H-TaSe2. In the case under investigation we have no direct evidence of DC.
However, the metastable subtle structural changes in the main crystalline
phase and the appearance of the higher order satellites $q2$ related to the
$q1$ are experimental indications in support to the thesis that the CDW in
LaPt2Si2 undergoes discommensuration, in qualitative agreement with the
2H-TaSe2 case. Moreover, there are clear satellite intensity fluctuations in
the full investigated temperature range (see figure 3), which can be
interpreted as due to coherent interference from the ordered commensurate
domains in the superlattice phase, in a similar way as it was done for
monolayer Kr/graphite thin film systems [18].

###  0.2  Bulk measurements

Magnetic susceptibility with magnetic field, equal to 0.5 T, 2 T, 5 T, and 7
T, applied along $a$-axis and $c$-axis and resistivity data with the current
running along $c$-axis on heating and cooling are displayed in figure 7. The
sharp drop in the resistivity curve below 2 K is due to the onset of the
superconducting transition, reportedly occurring at Tc = 1.22 K. The curve
between 2 K and 100 K is well fitted to a power law below 50 K and to a line
above, as usually occurs in normal metals. The value of the exponent as a
result of the power law fitting is $\alpha$ = 2.1 $\pm$ 0.1, which is
signature of normal Fermi liquid behavior and no strong electron-electron
interactions are in place. This could be indication of the fact that the
superconducting state established in LaPt2Si2 is of conventional nature,
however deeper analysis in needed to unambiguously support such a statement. A
sharp upturn in resistivity is observed in proximity of T2, across the first
order incommensurate to commensurate phase transition, and indicates a gap
opening at the Fermi surface. Usually in simple Peierls systems this
transition occurs between a metallic and an insulating phase. However,
metallic behavior below the transition suggests that the gap opening is
partial and does not take place over the entire Fermi surface. Interestingly,
the resistivity above T2 deviates from linearity bending downwards, with a
phenomenology similar to NbSe2 [19], in correspondence to the appearance of
the $q1$ satellites across the second order commensurate to incommensurate
phase transition. A possible interpretation for the negative curvature of the
resistivity is that it is probably caused by local fluctuations of the CDW
causing the opening of gaps on the Fermi surface which are too small to
macroscopically affect the transport properties in LaPt2Si2.

Figure 7: a) Resistivity curve as a function of temperature for LaPt2Si2
single crystal with the electric current applied along the $c$-axis, the solid
line is a guide to the eye to highlight the change of slope in the resistivity
curve b) Detail of the resistivity curve in the temperature range from 2 K to
100 K. The solid lines are fits to the power law and linear functions
respectively. c) Susceptibility curve for LaPt2Si2 with the magnetic field
flux lines oriented along the $a$-axis and d) $c$-axis with different values
of the field magnitude.

A first order drop in the value of magnetic susceptibility is observed in
proximity of T2, indicating reduction in the density of states due to partial
opening of a gap at Fermi surface. Differences between data with magnetic
field applied along $a$ and $c$ axes imply anisotropy of the gap. A small
anomaly in the susceptibility is observed in proximity of T1, probably
indicating the aforementioned small gap opening in correspondence to the
appearance of the $q1$ satellites. The published literature has not discussed
this second transition in the susceptibility data as well as the negative
curvature of the resistivity above T2, and attributes the first order
transition at T2 to the CDW transition. With the results reported in this
paper it is now possible to re-interpret these bulk measurements by placing
the CDW induced gap opening in correspondence to the small anomaly in the
susceptibility at T1 and by associating the sharp transition at T2 to a change
in the electronic states of LaPt2Si2 induced by the first order structural
transition from distorted tetragonal to orthorhombic crystal symmetry
associated with the second CDW transition $q2$ at T2.

## Discussion

The structural evolution and the temperature dependence of the development of
the density wave state in the CDW superconductor LaPt2Si2 was clarified with
synchrotron XRD and bulk characterization measurements. From our investigation
we concluded that the onset of the charge ordering is to be placed well above
room temperature. Phonon dispersion measurements would be relevant to clarify
this temperature dependence. On cooling, lattice distortions with multiple
propagation vectors occur and their temperature evolution can be followed
through the intensities of the corresponding satellites. In particular, four
temperature regimes can be identified:

  * •

T $>$ T1 = 230 K: diffuse scattering is present in the $ab$-plane while broad
satellites are already visible along the $c$-axis. The crystal structure of
LaPt2Si2 can be reliably refined with the tetragonal space group $P4/nmm$.

  * •

T2 $<$ T $<$ T1: satellites with propagation vector $q\prime 1$ = [0.36 0 0]
and $q\prime\prime 1$ = [0 0.36 0] become sharp and their intensity increases
on cooling. The crystal structure in LaPt2Si2 undergoes a second order
structural transition from the room temperature commensurate tetragonal phase
to a incommensurate distorted tetragonal phase.

  * •

T3 $<$ T $<$ T2: satellites with propagation vector $q\prime 2$ = [0.18 0.18
0.5] and $q\prime\prime 2$ = [0.18 -0.18 0.5] become sharp and their intensity
increases on cooling. The crystal structure in LaPt2Si2 undergoes a first
order structural transition at T = T2 = 110 K from the incommensurate
distorted tetragonal phase to a commensurate orthorhombic phase with space
group $Pmmn$.

  * •

T $<$ T3: satellites with propagation vector $q\prime 3$ = [0.3 0.3 0.25] and
$q\prime\prime 3$ = [-0.3 -0.3 0.25] appear and their intensity increases on
cooling from T3 = 60 K. The tetragonal structure with space group $P4/nmm$ is
restored after a first order transition.

A first CDW transition should be associated to the $q1$ satellites, the $q2$
satellites, associated to a second CDW transition, are related to the $q1$ as
higher order satellites and the $q3$ modulation of the lattice is established
as a consequence of the structural instabilities induced by the charge
modulation. First order transitions towards commensurate crystal structures
for the main cell occur in correspondence to the appearance of the $q2$ and
$q3$ superstructures. The long 3D phase coherence length for the $q1$ and $q2$
propagation vectors implies strong inter-planar interactions among the Pt1 and
Pt2 layers. This seems to indicate that 2 distinct CDW transitions occur: one
at T1 with propagation vector $q1$ in the Pt2 layer, and one at T2 with
propagation vector $q2$ in the Pt1 layer. The CDW induced ion displacement
indicates the presence of strong electron-phonon coupling, while the
metastable structural changes, the appearance of higher order satellites and
their intensity fluctuations can be indications of discommensuration of the
CDW in LaPt2Si2. This behavior is indeed similar to the behavior of other
systems that manifest CDW discommensuration and dislocation. However, no
direct evidence of DC is observed in this work, therefore, it is here
presented as a speculative conjecture. The temperature dependent behavior of
the crystal structure and the evolution of the CDW induced satellites
clarified in this study seem to point towards an unconventional character of
the LaPt2Si2 CDW states, with strong coupling between the Pt layers.

## Methods

The synchrotron X-ray diffraction measurements were performed on the P21.1
beamline [20] at the PETRA III synchrotron facility of the DESY national
research center (Deutsches Elektronen-Synchrotron). The sample was mounted on
a Displex cold finger cryostat, with a T-range 10 K – 320 K. The data were
acquired with a PILATUS3 X CdTe 2M detector over $360^{\circ}$ omega scans in
$0.1^{\circ}$ steps. Data at 41 temperatures were collected on controlled
heating in different steps depending on the range [10:10:50, 55:5:120,
130:10:150, 155:5:190, 200:10:300] K. The energy of the incoming photon beam
was selected to be $\approx$ 102 keV. In the low temperature region, there is
a difference of about 10 K between the temperature readout in the proximity of
the sample and the temperature set, therefore in the following we will always
refer to the sample temperature readout. The synchrotron XRD data were
collected with 3 different attenuation settings: atten0 for zero attenuation,
atten2 for intermediate attenuation, atten5 for maximum attenuation. Each
attenuation step is achieved through 0.1 mm Tl. Due to the strong attenuation
of the atten5 setting, not much information can be extracted from this
dataset, therefore it was disregarded.

Preliminary in-house XRD data were collected at the Arrhenius Laboratory in
Stockholm University. The resistivity and susceptibility measurements were
performed at the Physical Properties of Materials laboratory at the Paul
Scherrer Institute (PSI), Switzerland. The LaPt2Si2 sample was prepared using
arc melting of high purity La, Pt and Si at the Tata Institute of Fundamental
Research (TIFR), Mumbai [21].

All images involving crystal structure were made with the VESTA software [22],
the XRD data reduction and unit cell determination were carried out with the
software CrysAlisPro [23], the crystal structure refinement was performed with
the software SHELX [24], the data plots were produced with the software
IgorPro [25].

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## Acknowledgements

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association
HGF, for the provision of experimental facilities. Parts of this research were
carried out at PETRA III and we would like to sincerely thank O. Ivashko, M.
von Zimmermann and P. Glaevecke for assistance in using P21.1. Beamtime was
allocated for proposal 11013546. The bulk measurements were carried out at the
Laboratory for Multiscale Materials Experiments, Paul Scherrer Institut, in
Switzerland. The authors wish to thank the staff of PSI for the valuable
support provided during the measurements. The initial in-house XRD
characterization was carried out at the Stockholm University, Arrhenius
Laboratory (Department of Materials and Environmental Chemistry). The authors
wish to thank A. K. Inge for the valuable support provided during the
measurements, as well as A. Geresdi and N. Trnjanin, from the Chalmers
University of Technology, for the enlightening discussions around
unconventional superconductors. This research is funded by the Swedish
Foundation for Strategic Research (SSF) within the Swedish national graduate
school in neutron scattering (SwedNess).

## Author contributions statement

E.N. conceived the experiments. E.N., O.I., M.M., J.L., Y.M.K. conducted the
experiments. E.N, I.S. analyzed the results. The samples were synthesized by
Z.H. and A.T. who also conducted the initial sample characterizations. E.N.
and M.M made all the figures. E.N. created the first draft.

## Data availability statement

All the data of this work are available from the corresponding authors on
request.

Competing interests

The authors declare no competing interests.

The corresponding author is responsible for submitting a competing interests
statement on behalf of all authors of the paper.