# Coupling between colossal charge density wave ordering and magnetism in
Ho2Ir3Si5

Sitaram Ramakrishnan  Department of Quantum Matter, AdSE, Hiroshima
University, Higashi-Hiroshima 739-8530, Japan<EMAIL_ADDRESS>Jin-Ke Bao
Laboratory of Crystallography, University of Bayreuth, 95447 Bayreuth, Germany
Claudio Eisele  Laboratory of Crystallography, University of Bayreuth, 95447
Bayreuth, Germany  Bikash Patra  Department of Condensed Matter Physics and
Materials Science, Tata Institute of Fundamental Research, Mumbai 400005,
India  Minoru Nohara  Department of Quantum Matter, AdSE, Hiroshima
University, Higashi-Hiroshima 739-8530, Japan  Biplab Bag  Department of
Condensed Matter Physics and Materials Science, Tata Institute of Fundamental
Research, Mumbai 400005, India  Leila Noohinejad  P24, PETRA III, Deutsches
Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany  Martin
Tolkiehn  P24, PETRA III, Deutsches Elektronen-Synchrotron DESY, Notkestrasse
85, 22607 Hamburg, Germany  Carsten Paulmann  P24, PETRA III, Deutsches
Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany  Achim M.
Schaller  Laboratory of Crystallography, University of Bayreuth, 95447
Bayreuth, Germany  Toms Rekis  Laboratory of Crystallography, University of
Bayreuth, 95447 Bayreuth, Germany  Surya Rohith Kotla  Laboratory of
Crystallography, University of Bayreuth, 95447 Bayreuth, Germany  Andreas
Schönleber  Laboratory of Crystallography, University of Bayreuth, 95447
Bayreuth, Germany  Arumugam Thamizhavel  Department of Condensed Matter
Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai
400005, India  Bahadur Singh  Department of Condensed Matter Physics and
Materials Science, Tata Institute of Fundamental Research, Mumbai 400005,
India<EMAIL_ADDRESS>Srinivasan Ramakrishnan  Department of
Condensed Matter Physics and Materials Science, Tata Institute of Fundamental
Research, Mumbai 400005, India<EMAIL_ADDRESS>Sander van Smaalen
Laboratory of Crystallography, University of Bayreuth, 95447 Bayreuth, Germany
<EMAIL_ADDRESS>
###### Abstract

Ho2Ir3Si5 belongs to the family of three-dimensional (3D) $R_{2}$Ir3Si5 ($R$ =
Lu, Er and Ho) compounds that exhibit a colossal first-order charge density
wave (CDW) transition where there is a strong orthorhombic-to-triclinic
distortion of the lattice accompanied by superlattice reflections. The
analysis by single-crystal X-ray diffraction (SXRD) has revealed that the Ir-
Ir zigzag chains along c are responsible for the CDW in all three compounds.
The replacement of the rare earth element from non-magnetic Lu to magnetic Er
or Ho lowers $T_{CDW}$, where $T_{CDWLu}$ = 200 K, $T_{CDWEr}$ = 150 K and
$T_{CDWHo}$ = 90 K. Out of the three compounds, Ho2Ir3Si5 is the only system
where second-order superlattice reflections could be observed, indicative of
an anharmonic shape of the modulation wave. The CDW transition is observed as
anomalies in the temperature dependencies of the specific heat, electrical
conductivity and magnetic susceptibility, which includes a large hysteresis of
90 to 130 K for all measured properties, thus corroborating the SXRD
measurements. Similar to previously reported Er2Ir3Si5, there appears to be a
coupling between CDW and magnetism such that the Ho3+ magnetic moments are
influenced by the CDW transition, even in the paramagnetic state. Moreover,
earlier investigations on polycrystalline material revealed antiferromagnetic
(AFM) ordering at $T_{N}$ = 5.1 K, whereas AFM order is suppressed and only
the CDW is present in our highly ordered single-crystal. First-principles
calculations predict Ho2Ir3Si5 to be a metal with coexisting electron and hole
pockets at the Fermi level. The Ho and Ir atoms have spherically symmetric
metallic-type charge density distributions that are prone to CDW distortion.
Phonon calculations affirm that the Ir atoms are primarily responsible for the
CDW distortion, which is in agreement with the experiment.

[Second University] Department of Physics, Materials Genome Institute and
International Center for Quantum and Molecular Structures, Shanghai
University, Shanghai 200444, People’s Republic of China [Second University]
Amity Institute of Applied Sciences, Amity University Jharkhand, India-834001
[Second University] Mineralogisch-Petrographisches Institut, Universität
Hamburg, 20146 Hamburg, Germany [Second University] IISER, Dr. Homi Bhabha
Road, Pashon, Pune-411008, India

##  1 Introduction

Structural phase transitions, such as the development of charge density waves
(CDWs), have attracted continued interest from condensed-matter physicists and
chemists. Originally, the manifestation of a CDW was proposed to originate in
Fermi surface nesting (FSN), as it is present in quasi-one-dimensional metals
(1D) 1. More recently, alternative mechanisms were proposed that explain, for
example, the formation of CDWs by $q$-dependent electron-phonon coupling (EPC)
in three-dimensional (3D) CDW compounds. Examples of 3D metals with CDWs
include $\alpha$-Uranium 2, CuV2S4 3, 4, 5, 6, La3Co4Sn13 7, 8, 9, $R$Te3 ($R$
= La, Sm, Gd, Tb, Dy, Ho, Er, Tm) 10, 11, 12, $R$Te2 ($R$ = La, Ce) 13, 14,
$R$5Ir4Si10 ($R$ = Dy, Ho, Er, Yb, Lu) 15, Sm2Ru3Ge5 16, EuAl4 17, 18, 19, 20,
21 and CuIr2-xCrxTe4 22. In several of these compounds a coexistence and
competition exists between the CDW and antiferromagnetic (AFM) order or
superconductivity (SC). In the family of compounds $R$NiC2 ($R$ = Ce, Pr, Nd,
Sm, Gd, Tb, Dy, Ho, Er, Tm) 23, 24, 25, 26, 27, 28, 29, the CDW competes with
AFM. In the case of SmNiC2 ferromagnetic (FM) order completely destroys the
CDW 24, 25. On the other hand, Kolincio et al. 29 have established the
coexistence of a CDW and field-induced FM order in TmNiC2, suggesting strong
coupling of the rare-earth spins to the CDW in these rare-earth materials.
However, the magnetic susceptibility of the paramagnetic regime does not
exhibit anomalies at the CDW transitions.

Er2Ir3Si5 differs from most magnetic CDW compounds, in that the magnetic
susceptibility of the paramagnetic state exhibits an anomaly at the CDW
transition 30. Presently, we report a similar effect for Ho2Ir3Si5. On the
other hand, the CDW of non-magnetic Lu2Ir3Si5 coexists with SC at low
temperatures 31.

Investigations of the CDW in Lu2Ir3Si5 has been extensively done 32, 33, 34,
35. Studies by transmission electron microscopy (TEM) reported the modulation
wave vector as q = $\sigma(\bar{1}21)$ with $\sigma$ = 0.23–0.25 around 200 K
35. Recently, the modulated crystal structure of Lu2Ir3Si5 was reported as
similar to that of Er2Ir3Si5 36. In both systems it was elucidated that the
CDW resides on the zigzag chains of Ir atoms along c. Although the rare earth
element is not directly involved in the stabilization of the CDW, we have
earlier proposed that it would indirectly influence the CDW through its size
(atomic radius). Er has a larger atomic radius than to Lu has (2.26 vs 2.17
Å),37 while the transition occurs 50 K lower in Er2Ir3Si5 than in Lu2Ir3Si5.
Here, we find that there is no such simple relationship between atomic radius
and $T_{CDW}$. The atomic radii of Ho and Er are equal, but $T_{CDW}$ = 90 K
of Ho2Ir3Si5 is much lower than $T_{CDW}$ = 150 K of Er2Ir3Si5.

From SXRD we found that Ho2Ir3Si5 undergoes a large distortion of the lattice
where the symmetry is lowered from orthorhombic ($Ibam$) to triclinic
($I\bar{1}$) below the transition temperature $T_{CDW}$. Simultaneously,
superlattice reflections appear at incommensurate positions, with values of
$\mathbf{q}$ = $[0.2494(2),\>0.4978(2),\>0.2488(2)]$ at 70 K. The CDW is
supported by zigzag chains of Ir atoms in all three compounds $R_{2}$Ir3Si5.
Ho2Ir3Si5 differs from Lu2Ir3Si5 and Er2Ir3Si5 by the presence of second-order
superlattice reflections in its SXRD, indicative of anharmonic contributions
to the displacive modulation wave.

We present for Ho2Ir3Si5 the temperature dependencies of the electrical
resistivity $\rho(T)$, the specific heat $C_{p}(T)$ and the magnetic
susceptibility $\chi(T)$. They show anomalies in agreement with a first-order
CDW transition with a hysteresis of about 40 K. Due to the strain imposed by
the transition onto the crystal, it cracks which can be seen clearly from the
resistivity measurements. Such cracking of crystals has also been observed in
other CDW compounds, like BaFe2Al9 38. From the magnetic susceptibility we
observed that there is an influence of the CDW on the effective magnetic
moment of Ho3+, which is similar to what was reported in Er2Ir3Si5 30. Such an
effect of the CDW on the rare earth spin in the paramagnetic state is not
commonly seen in rare earth compounds. Earlier studies have shown that the
disorder in the polycrystalline material allows long-range AFM order to
develop at low temperatures 32, 33. However, in the present high-quality
single crystal, long-range AFM order is absent and only the CDW is observed.
Here, we elucidate the differences and similarities in the incommensurate
structure of Ho2Ir3Si5 by comparing with Lu2Ir3Si5 and Er2Ir3Si5. Furthermore,
we explore the coupling of the CDW and magnetism of Ho3+ spins and absence of
long-range AFM order.

##  2 Experimental and computational details

###  2.1 Crystal growth and characterization

Figure 1: Crystal growth in a tetra arc furnace by the Czochralski method. A
bar was obtained of about 50 mm length and 15 mm in diameter.

A single crystal of Ho2Ir3Si5 has been grown by the Czochralski method in a
tetra-arc furnace as shown in Fig. 1. We have chosen the Czochralski over flux
growth or any other technique because the individual elements have high
melting points. Furthermore, iridium metal does not dissolve in most of the
typical fluxes. To start with, high purity individual elements of Ho : Ir : Si
(99.99% for Ho and Ir, and 99.999% for Si) were taken in the stoichiometric
ratio 2 : 3 : 5, amounting to about 8 to 9 g, and melted repeatedly to ensure
its homogeneity. Next, a seed crystal was cut from this polycrystalline ingot
for the purpose of crystal growth. The polycrystalline seed was gently
inserted into the molten solution and initially pulled at a rapid speed of
about 80 mm/h. The temperature of the melt was adjusted such that a necking is
formed and then we employed a pulling speed of about 10 mm/h throughout the
growth process. About 50 mm long ingot was pulled. Energy-dispersive X-ray
spectroscopy (EDX) was used to verify the chemical composition.

###  2.2 Single-crystal X-ray diffraction (SXRD): data collection and data
processing.

Table 1: Crystallographic data of crystal A of Ho2Ir3Si5 at 200 K (periodic phase) and 70 K (CDW phase). Temperature (K) | 200 | 70
---|---|---
Crystal system | Orthorhombic | Triclinic
Space/Superspace group | $Ibam$ | $I\bar{1}(\sigma_{1}\>\sigma_{2}\>\sigma_{3})0$
Space/Superspace group No. 39 | 72 | 2.1.1.1
$a$ (Å) | 9.9023(4) | 9.8356(5)
$b$ (Å) | 11.3747(3) | 11.4902(4)
$c$ (Å) | 5.7745(3) | 5.7304(3)
$\alpha$ (deg) | 90 | 89.983(3)
$\beta$ (deg) | 90 | 91.772(2)
$\gamma$ (deg) | 90 | 89.975(1)
Volume (Å3) | 650.42(5) | 647.32(5)
Wave vector q | - | (0.2494(2), 0.4978(2), 0.2488(2))
$Z$ | 4 | 4
Wavelength (Å) | 0.50000 | 0.50000
Detector distance (mm) | 110 | 110
$2\theta$-offset (deg) | 0 | 0
$\chi$-offset (deg) | -60 | -60
Rotation per image (deg) | 1 | 1
$(\sin(\theta)/\lambda)_{max}$ (Å-1) | 0.683589 | 0.684039
Absorption, $\mu$ (mm-1) | 34.823 | 34.990
Tmin, Tmax | 0.0220, 0.0501 | 0.0230, 0.0501
Criterion of observability | $I>1.5\sigma(I)$ | $I>1.5\sigma(I)$
Number of $(m=0)$ reflections |  |
measured | 3765 | 2706
unique (obs/all) | 451/470 | 1298/1474
Number of $(m=1)$ reflections |  |
measured | - | 12770
unique (obs/all) | - | 2779/6454
Number of $(m=2)$ reflections |  |
measured | - | 12725
unique (obs/all) | - | 714/3263
$R_{int}$ $(m=0)$ (obs/all) | 0.0487/0.0487 | 0.0287/0.0288
$R_{int}$ $(m=1)$ (obs/all) | - | 0.0828/0.1051
$R_{int}$ $(m=2)$ (obs/all) | - | 0.0967/0.1833
No. of parameters | 31 | 147
$R_{F}$ $(m=0)$ (obs) | 0.0296 | 0.0578
$R_{F}$ $(m=1)$ (obs) | - | 0.0798
$R_{F}$ $(m=2)$ (obs) | - | 0.2148
$wR_{F}$ $(m=0)$ (all) | 0.0380 | 0.0717
$wR_{F}$ $(m=1)$ (all) | - | 0.0973
$wR_{F}$ $(m=2)$ (all) | - | 0.3748
$wR_{F}$ all (all) | 0.0380 | 0.0879
GoF (obs/all) | 1.83/1.33 | 1.69/1.23
$\Delta\rho_{min}$, $\Delta\rho_{max}$ (e Å-3) | -3.00, 2.94 | -14.56, 14.53

Small pieces of single-crystalline Ho2Ir3Si5 were obtained by crushing a large
single crystal, from which crystal A of dimensions 0.15$\times$0.07$\times$0.1
mm3 was selected for a single-crystal X-ray diffraction (SXRD) experiment at
beamline P24 of PETRA-III Extension at DESY in Hamburg, Germany. SXRD was
measured at station EH2 of beamline P24, employing radiation of a wavelength
of $\lambda_{P24}$ = 0.50000 Å. For further details regarding data collection
refer to the supporting information 40.

The EVAL15 software suite 41 was used for processing the SXRD data. SADABS 42
was used for scaling and absorption correction, with Laue symmetry $mmm$ for
the data in the periodic phase and $\bar{1}$ for the CDW phase. As the crystal
structure in the CDW phase is incommensurately modulated we had to use the
superspace approach 43, 44, 39 to index and integrate the data. Section S2 in
the supporting information 40 provides further details. The resulting
reflection file was imported into JANA2006 45, 46. Table 1 gives the
crystallographic information at 200 K (periodic phase) and at 70 K
(incommensurate phase). The crystallographic data at other temperatures and
details regarding SXRD data processing are given in the Supporting
Information.40

###  2.3 Physical properties

A commercial superconducting quantum interference device (SQUID) magnetometer
(MPMS5, Quantum Design, USA) was used to measure dc magnetic susceptibility in
a field of 100 mT as a function of temperature from 2 to 300 K. The electrical
resistivity between 1.8 and 300 K was measured by the standard dc four probe
technique on a commercial physical property measurement system (PPMS, Quantum
Design, USA). The specific heat data were measured both on PPMS as well as
using a commercial differential scanning calorimeter (DSC) setup.

###  2.4 Density functional theory calculations

Density functional theory (DFT) based calculations were performed using the
projector augmented wave (PAW) 47 method as implemented in the Vienna ab
initio simulation package (VASP).48 Exchange-correlation effects were included
using the Perdew-Burke-Ernzerhof (PBE) 49 version of the generalized gradient
approximation. A $8\times 10\times 12$ $\Gamma$-centered $k$-mesh was used for
the Brillouin zone (BZ) sampling with an energy cut-off of $380$ eV for the
plane-wave basis set. Spin-orbit coupling (SOC) effects were taken into
account self-consistently to consider the relativistic effects. We employed
Ho3+ potential by considering the remaining 4f electrons as core electrons.
Experimental lattice parameters were used, while the internal atomic positions
were relaxed until the residual forces on each atom were less than 0.0001
eV/Å. The VESTA50 program was used to visualize the charge density
distributions. The phonon spectrum was obtained using a $2\times 2\times 2$
supercell following the frozen phonon method as implemented in the phonopy 51
package.

##  3 Results and discussion

###  3.1 Analysis of the CDW structure

Unlike Er2Ir3Si5,30 the manifestation of the CDW appears to be sluggish in
Ho2Ir3Si5. The crystal was initially cooled from room temperature down to 70
K, where it remained in the orthorhombic phase, despite $T_{CDW}$ being about
90 K according to the physical property measurements.

It is possible that the crystal was cooled too rapidly, thereby not allowing
it to settle in the CDW phase, and resulting in an undercooled state. Upon
further lowering of the temperature to 20 K, we observed the coexistence of
orthorhombic and CDW phases, indicative of a first-order transition (Figs. 2
and 3). The CDW phase is characterized by superlattice reflections at
positions $\mathbf{q}$ = $[0.2496(3),\>0.4987(3),\>0.2493(3)]$, accompanied by
a large monoclinic distortion of the lattice ($\beta=91.784(3)$ deg), similar
to $R_{2}$Ir3Si5 ($R$ = Lu, Er) (details are given in the Supporting
Information40).

The severity of the distortions induces a colossal strain in the crystal,
causing the diffraction spots in SXRD to become broad and elongated [Figs.
2(b) and 2(c)], resulting in many partially overlapped reflections which
cannot be used for structural analysis. Sometimes the strain is too much for
the crystal to handle, such that it physically shatters, rendering it unusable
for further investigations. Such destructive behaviour of the CDW has also
been reported for BaFe2Al9 38.

Another important difference to Lu2Ir3Si5 and Er2Ir3Si5 is that we observe in
SXRD on Ho2Ir3Si5 second-order superlattice reflections $(m=2)$ in addition to
first-order superlattice reflections $(m=1)$ [Figs. 2(e) and 2(f)]. The
orthorhombic phase disappears after the crystal is heated to 50 K, and only
the CDW phase remains at temperatures 50–130 K (Figs. 2(c) and 3). Further
heating to 150 and 200 K causes the crystal to enter the orthorhombic phase
and the CDW phase disappears [Fig. 2(a)]. Similar to Lu2Ir3Si5 and Er2Ir3Si5,
there is a lowering of the point symmetry from orthorhombic to triclinic which
causes the crystal to be twinned with four orientations 52.

Figure 2: (a), (b), (c) The $h\,0\,l$ plane, and (d), (e), (f) the $h\,k\,h$
plane reconstructed from measured SXRD data. (a), (d) are for SXRD on the
periodic phase at 200 K. (b), (e) show the intermediate phase (undercooled
state), where threefold splitting of the reflections can be observed at 20 K,
indicating coexistence of CDW and periodic phases. (c), (f) are for SXRD on
the CDW phase at 50 K, showing groups of two instead of groups of three split
reflections. The degree of splitting increases with the value of $h$. Panels
(e) and (f) show satellite reflections of order $m=2$.

Figure 3 shows the temperature dependence of the lattice parameters. One can
observe the clear distortion of the lattice upon entering the CDW state,
including an expansion of $b$ and contractions of $a$ and $c$. Furthermore,
the change in lattice type at $T_{CDW}$ appears discontinuous, in agreement
with the first-order character of the phase transition. Barring 20 K and 130 K
as 20 K is an undercooled mixed state and 130 K is on the onset of $T_{CDW}$
warming and therefore unreliable, it can be inferred from Fig. 3(d) that the
modulation wave vector q decreases with temperature.

Figure 3: Temperature dependence of (a) the lattice parameters $a$, $b$ and
$c$ relative to their values at $T=200$ K: $a(200)=9.9023(4)$,
$b(200)=11.3747(3)$ and $c(200)=5.7745(3)$ Å; (b) the lattice parameters
$\alpha$, $\beta$ and $\gamma$; (c) the volume of the unit cell; and (d)
components of the modulation wave vector q. Orange and blue symbols refer to
data obtained during heating and cooling of the crystal, respectively. The
undercooled state at 20 K has coexisting CDW and periodic phases.

The presence of second-order satellite reflections required refinement of the
crystal structure with higher order harmonics for the modulation functions.
Table 2 provides a comparison of the models to see which provides the best fit
to the SXRD data in the CDW phase.

Table 2: Quality of the fit to the SXRD data at 70 K for three structure models for the CDW phase. See text for models A, B and C. The number of unique reflections is 1298/1474 for obs/all main reflections $(m=0)$, 2779/6454 for obs/all $m=1$ satellites, and 714/3263 for obs/all $m=2$ satellites. Criterion of observability is $I>1.5\sigma(I)$. Models | A | B | C
---|---|---|---
No. of parameters | 147 | 177 | 177
$R_{F}$ $(m=0)$ (obs) | 0.0578 | 0.0577 | 0.0535
$R_{F}$ $(m=1)$ (obs) | 0.0798 | 0.0796 | 0.0672
$R_{F}$ $(m=2)$ (obs) | 0.2148 | 0.2119 | 0.0894
$wR_{F}$ $(m=0)$ (all) | 0.0717 | 0.0717 | 0.0695
$wR_{F}$ $(m=1)$ (all) | 0.0973 | 0.0972 | 0.0869
$wR_{F}$ $(m=2)$ (all) | 0.3748 | 0.3721 | 0.1319
$wR_{F}$ all (all) | 0.0879 | 0.0877 | 0.0738
GoF (obs/all) | 1.69/1.23 | 1.69/1.23 | 1.52/1.03
$\Delta\rho_{min}$, $\Delta\rho_{max}$ (e Å-3) | -14.56, 14.53 | -14.52, 14.69 | -8.73, 6.14

####  3.1.1 Model A

Here, up to second-order harmonics have been applied to holmium and iridium
atoms, whereas silicon atoms have only first-order harmonics (Eq. S6 in 40).
This resulted in fewer parameters and minimal changes to the results as
compared to models B and C (Table 2). This model has been selected to describe
the crystal structure in the CDW phase.

####  3.1.2 Model B

In this model up to second-order displacement modulation parameters have been
applied to all atoms and then refined. The refinement resulted in a fit to the
SXRD data that is almost identical to that of Model A. However, the second-
order modulation amplitudes of the silicon atoms appear to have large standard
uncertainties (s.u.’s). This is probably due to the poor scattering power of
silicon as compared to holmium or iridium. Together, these features imply that
the additional 30 parameters in model B do not lead to a significant
improvement and model B is not selected.

####  3.1.3 Model C

Model C builds upon model A and model B in order to try solve the issue with
the high value of $R_{F}$ for the second-order satellites. Up to third-order
harmonics are used for the modulation functions of the holmium and iridium
atoms, while employing only first-order harmonics for silicon atoms, as
higher-order harmonics have been discarded for Si on the basis of models A and
B. As a rule of thumb the highest harmonics in the modulation wave should not
exceed the highest order of observed satellite reflections, the latter which
is two in the present SXRD experiment. At first glance from Table 2 we see a
huge improvement of the fit to the second-order satellites upon introduction
of third-order harmonics. However, all these parameters have a s.u. that is
larger than the refined value itself. The latter indicate that the system
would have third-order satellite reflections with intensities larger than its
first- and second-order satellites. As we did not observe third-order
satellites, the refined values of the third-order harmonics appear to be
unreliable, and model A is chosen in favor of model C.

###  3.2 Location of the CDW

Table S8 in the Supporting Information 40 shows the atomic coordinates at 200
K and 70 K (warming). The six crystallographically independent atoms, Ho1,
Ir1, Ir2, Si1, Si2, and Si3, of the $Ibam$ structure at 200 K split into the
Ho1a, Ho1b, Ir1a, Ir1b, Ir2, Si1, Si2a, Si2b, Si3a, Si3b atoms of the
triclinic structure at 70 K. Figure 4 shows the crystal structure projected
onto the a-c plane at 200 K and 70 K (average structure at 70 K). Figure 4(b)
appears skew compared to Fig. 4(a) due to $\beta>90$ deg in the CDW phase. As
the modulation wave vector is close to (1/4, 1/2, 1/4), Fig. 4(c) shows a
$4\times 2\times 4$ superstructure approximation, where one can see zigzag
chains of Ir1a-Ir1b along c. Tables S9 and S10 in the Supporting Information
40 show the refined modulation amplitudes.

Figure 4: Projection onto the $(\mathbf{a},\mathbf{c})$-plane of the crystal
structures of Ho2Ir3Si5 for (a) 200 K and (b,c) 70 K. Panel (c) shows the
$4\times 2\times 4$ superstructure approximation. Large purple spheres
correspond to Ho atoms; green spheres of intermediate size correspond to Ir
atoms; small yellow spheres are Si atoms. Dashed lines give the distances in
the basic structure, with values of $3.390\,(3)$ and $3.728\,(3)$ Å.

Analysis of the modulation amplitudes and distances revealed that Ho2Ir3Si5
follows a similar pattern as $R_{2}$Ir3Si5 (Lu, Er), 36, 30 such that the CDW
resides on zigzag chains of Ir1a-Ir1b atoms, as they have the shortest metal-
metal distances and exhibit the largest modulation of these distances (Fig.
5). The lattice distortion results in alternating short and long Ir1a–Ir1b
distances, where the shorter distance is the most affected by the modulation.
The formation of dimers on the Ir1a-Ir1b zigzag chains along c is responsible
for the formation of the CDW.

Figure 5: $t$-Plot of the interatomic distances between atoms Ir1a and Ir1b
$(x,y,z)$ and between Ir1a and Ir1b at $(x,y,z-1)$ for the crystal in the CDW
phase at $T=70$ K.

Table 3 provides a comparison of essential structural parameters between the
three compounds $R_{2}$Ir3Si5. Our earlier proposal, that $T_{CDW}$ would be
inversely proportional to the atomic radius of the rare earth element, is not
confirmed by Ho2Ir3Si5. The atomic radius of Ho is comparable to that of Er,
however $T_{CDW}$ of Ho2Ir3Si5 is much lower than that of Er2Ir3Si5. The
variation of distances Ir1a–Ir1b is more or less similar in all three
compounds.

Table 3: Crystal data of the CDW phases of Lu2Ir3Si5 31, 36, Er2Ir3Si5 30 and
Ho2Ir3Si5 (present results).

Compound | Lu2Ir3Si5 | Er2Ir3Si5 | Ho2Ir3Si5
---|---|---|---
Atomic radius of $R$ (Å) 37 | 2.17 | 2.26 | 2.26
$T_{\mathrm{CDW}}$ (K) | 202–231 | 150–166 | 90–130
$T$ (K) | 60 | 75 | 70
$a$ (Å) | 9.8182(3) | 9.8494(3) | 9.8356(5)
$b$ (Å) | 11.4093(3) | 11.4863(3) | 11.4902(4)
$c$ (Å) | 5.6835(2) | 5.7268(2) | 5.7304(3)
$\alpha$ (deg) | 90.001(2) | 90.079(1) | 89.983(3)
$\beta$ (deg) | 91.945(2) | 91.695(2) | 91.772(2)
$\gamma$ (deg) | 90.018(2) | 90.051(1) | 89.975(1)
$V$ (Å3) | 636.34(3) | 647.60(5) | 647.32(5)
$q_{x}$ | 0.2499(3) | 0.2495(2) | 0.2494(2)
$q_{y}$ | 0.4843(4) | 0.4973(1) | 0.4978(2)
$q_{z}$ | 0.2386(2) | 0.2483(1) | 0.2488(2)
Distance Ir1a–Ir1b _a_ |  |
max (Å) | 3.801(1) | 3.818(2) | 3.801(3)
min (Å) | 3.711(1) | 3.714(2) | 3.719(3)
avg (Å) | 3.755(1) | 3.764(2) | 3.763(3)
Distance Ir1a–Ir1b _b_ |  |
max (Å) | 3.761(1) | 3.782(2) | 3.728(3)
min (Å) | 3.002(1) | 3.008(2) | 3.053(3)
avg (Å) | 3.385(1) | 3.398(2) | 3.390(3)

_a_ Symmetry code for Ir1b $(x,y,z)$; given are the maximum (max), minimum
(min) and average (avg) distances. _b_ Symmetry code for Ir1b $(x,y,z-1)$.

###  3.3 Electronic structure and phonons

The band structure of Ho2Ir3Si5 is shown in Fig. 6(a) for several high-
symmetry directions in the primitive BZ given in Fig. 6(b).

Figure 6: (a) Bulk electronic band structure of the orthorhombic phase of
Ho2Ir3Si5 along various high-symmetry directions in the primitive Brillouin
zone. (b) Brillouin zone with high-symmetry points. (c) Total (highlighted
grey color) and orbital projected density of states (green, red, and blue
lines). (d) and (e) Electronic charge density distributions on planes
containing Ho and Ir atoms, respectively. (f) Phonon band structure of the
orthorhombic phase, as calculated with smearing parameter $\sigma=0.05$ eV.
(g) Phonon density of states. The imaginary frequencies are represented by
negative values and are mainly associated with Ir atoms. (h) Expanded view of
the phonon band structure along the $V$–$R$–$Y$ directions. The absence of
imaginary frequencies for smearing parameter $\sigma=0.5$ eV indicates the
stability of the orthorhombic structure towards higher temperatures.

It is metallic such that both electron and hole pockets exist at the Fermi
level. To resolve the contributions of electronic states near the Fermi
energy, we show the atomic orbital projected density of states (PDOS) in Fig.
6(c). The Ir states are dominant near $E_{F}$, whereas the Ho and Si states
have lower weight than those of Ir atoms. To understand the charge density
distributions of these atoms, we have shown the charge density on two
different planes in the primitive unit cell, which contain Ho and Ir atoms,
respectively [Figs. 6(d) and 6(e)]. The spherical charge distribution of these
atoms is reminiscent of the metallic type ionic environment and suggests that
these atoms are more likely to undergo CDW modulations to find a low energy
state.53

Figure 6(f) shows the phonon spectrum along various high-symmetry directions
in the BZ. The system is dynamically unstable with Kohn-type 54 soft modes at
$Z$, $R$ and in between $Y$ and $U$ points. The phonon PDOS [Fig. 6(g)] shows
that Ho and Ir atoms contribute to the low-frequency phonon modes, whereas
high-frequency phonon modes are dominated by Si atoms. The imaginary
frequencies are inherent to the Ir atoms, confirming that the CDW is
associated mainly with the Ir atoms. The lowest value of negative frequency is
found near the reciprocal point (0.25, 0.50, 0.25), which is consistent with
the incommensurate wave vector $\mathbf{q}$ found in the SXRD experiment.
Figure 6(h) demonstrates the evolution of the soft phonon mode as a function
of the smearing parameter $\sigma$, which represents the electronic
temperature in our calculations. The soft phonon mode disappears upon
increasing the smearing parameter $\sigma$ from 0.05 eV to 0.5 eV. This
particular dependence of the phonon frequencies on the smearing parameter
(electronic temperature) indicates that the orthorhombic structure is stable
only at higher temperatures as seen in our experiments.

###  3.4 Electrical resistivity

Figure 7: Temperature dependence of the electrical resistivity $\rho(T)$ of
Ho2Ir3Si5. The pink and green regions indicate the periodic and CDW phases,
respectively. A large hysteresis of 40 K is clearly seen. The crystal
shattered on heating to above 130 K.

From Fig. 7 we infer that, upon cooling the crystal, there is a sharp upward
turn of the electrical resistivity at 90 K, which signifies the opening of a
gap in the electronic density of states over a major fraction of the Fermi
surface, in agreement with the formation of a charge density wave (CDW) at
this temperature. The metallic nature of the crystal after the CDW transition
indicates a partial opening of the gap at the Fermi surface. Upon heating,
$\rho(T)$ shows a sharp anomaly at 130 K, thereby establishing a huge
hysteresis of 90–130 K. This large hysteresis and the sharpness of the
transition signify that it is a first-order transition much like the one seen
in $R_{2}$Ir3Si5 ($R$ = Lu, Er) 31, 30. Such a large hysteresis is uncommon
for CDW transitions. Especially for incommensurate CDWs the transition usually
is of second order, e.g. as found in the canonical CDW system NbSe3 55.

All measurements were made using virgin samples. After cooling and then
heating through the phase transition, micro-cracks develop due to the strain
induced by the transition. This feature is visible in the electrical
resistivity through a lower resistivity of virgin samples than of samples that
have gone through a cooling/heating cycle (Fig. 7). Similar behavior has been
observed for ceramic material of $R_{2}$Ir3Si5 ($R$ = Lu, Er) 32, 33. It was
explained by variations of pinning of the CDW in those systems. However, it is
possible that the large lattice distortions accompanying the CDW transition
could be responsible for the formation of micro-cracks. This leads to a
reduction of sizes of mosaic blocks, and an increase in texture, which, in
turn, would cause an increase of the electrical resistance at each thermal
cycle. The presence of micro-cracks is supported by the observation that the
transition temperatures and hysteresis are the same in each thermal cycle. On
the other hand, the increase of pinning centers would lead to a lowering of
the CDW transition temperature, which is not observed here. Just as in the
case of $R_{2}$Ir3Si5 ($R$ = Lu, Er) 36, 30, we observe that the electrical
resistivity exhibits a $T^{2}$ dependence below $T$ = 50 K, which is up to
much higher temperatures than $\sim$10 K for a Fermi liquid, implying dominant
contributions from short-range magnetic fluctuations of Ho spins in the
absence of long-range magnetic ordering.

###  3.5 Magnetic susceptibility

The temperature-dependent magnetic susceptibility, $\chi(T)$, of Ho2Ir3Si5 was
measured on a single crystal from the same rod as used for the other bulk
measurements. Measurements were performed with a commercial superconducting
quantum interference device (SQUID) magnetometer (MPMS 7, Quantum Design, USA)
employing a magnetic field of 0.1 T along $\mathbf{c}$, during cooling and
heating between 2 and 300 K (Fig. 8).

Figure 8: Temperature dependence of the inverse magnetic susceptibility
($1/\chi(T)$) of Ho2Ir3Si5. The upper inset shows an expanded view of
$1/\chi(T)$ versus temperature. The lower inset shows an expanded view
$\chi(T)$ versus temperature.

The most interesting feature of the temperature-dependent magnetic
susceptibility is the noticeable increase of $\chi$(T) at 90(1) K upon cooling
through the CDW transition. The observed effect cannot be explained by a
change of Pauli susceptibility at the transition, since the estimated
magnitude of the Pauli susceptibility is at least two orders of magnitude
smaller than the observed change in the susceptibility. Furthermore, one would
expect a smaller Pauli susceptibility in the CDW phase, whereas we observe an
increase of the susceptibility when cooling through the transition. The same
transition is found at 130(1) K upon heating. The observed hysteresis is in
good agreement with the hysteresis in the electrical resistivity and DSC
measurements. A Curie-Weiss fit to the data at 140–300 K results in a Curie
constant of $C$ = 31.78(1) emu/mol K and an antiferromagnetic Weiss
temperature of $\theta$ = -2.6(1) K. A Curie-Weiss fit to the low-temperature
paramagnetic regime 40–90 K results in $C$ = 32.4(1) emu/mol K and $\theta$ =
-6.3(1) K. The different Curie constants correspond to different effective
magnetic moments on Ho3+ of 11.27 $\mu_{B}$ and 11.38 $\mu_{B}$, respectively.
These values are slightly higher than the free ion magnetic moment of Ho3+
ions.

So we see that Ho2Ir3Si5 joins our earlier studied Er2Ir3Si5 30 as yet another
exceptional case in showing an effect of the CDW transition on the magnetic
susceptibility. Usually, compounds containing magnetic rare-earth elements do
not show any anomaly in the paramagnetic susceptibility at the high-
temperature CDW transitions. This is true for many of the magnetic CDW
compounds which are mentioned in the introduction. For example, the
paramagnetic susceptibility of Ho5Ir4Si10 does not show any anomalies at its
CDW transition 56, 57. The coexistence of antiferromagnetic order ($T_{N}$ = 2
K) and CDW in Ho5Ir4Si10 might be related to the presence of weakly coupled
$4f$ electrons of Ho3+ ions 15, while for Ho2Ir3Si5 the reduced magnitude of
the magnetic moments and strong coupling suggest influence of the $4f$
electrons in the CDW transition and vice versa. It is not clear, whether the
small but distinct change of Ho3+ moment (11.27 $\mu_{B}$ vs 11.38 $\mu_{B}$)
across the CDW transition in a single crystal of Ho2Ir3Si5 could be
responsible for the absence of magnetic ordering of Ho3+ moments in the
crystal down to 2 K.

###  3.6 Specific heat

Figure 9: Temperature dependence of the specific heat $C_{p}$ from 2 to 250 K
using PPMS. The inset provides an enlarged view of the CDW transition where
$\Delta C_{p}$ = 60.5(1) J/(mol K). Figure 10: Temperature dependence of the
excess specific heat $\Delta C_{p}$, obtained by subtracting a smooth baseline
from the DSC signal. Blue and red circles refer to cooling and warming data.
Clear peaks are observed in both cooling and warming with a hysteresis of
about 40 K.

The specific heat ($C_{p}(T)$) of Ho2Ir3Si5 was measured by the thermal
relaxation method, using a physical property measuring system (PPMS, Quantum
Design, USA). Data obtained during heating of a single crystal of 10.5 mg from
2 to 250 K exhibit a sharp peak at the temperature of the CDW transition at
131.1 (1) K (Fig. 9). We are unable to find a peak in $C_{p}(T)$ while cooling
the crystal. Similar behavior was noted earlier for a crystal of Lu2Ir3Si5 31.
This feature could be the result of the specific method of measurement
employed in the PPMS instrument. In a second experiment, differential scanning
calorimetry (DSC) was measured from 80 to 150 K on the same single crystal of
Ho2Ir3Si5 (Fig. 10). The DSC data show clear peaks at different temperatures
in the heating and cooling runs, that appear at similar temperatures as are
found for the electrical resistivity and magnetic susceptibility, and thus
confirm the first-order character of the phase transition. The peaks appear of
similar width as for Lu2Ir3Si5 31, while the DSC peaks are much sharper for
Er2Ir3Si5 30. Since other features, like $C_{p}(T)$ and $\rho(T)$, exhibit
very sharp anomalies at the CDW transition, the broadened features in the DSC
signal might be related to the rate of change of temperature in this
experiment in conjunction with the sluggish character of the transition.

The lattice contribution to the specific heat was determined from a fit to the
data far away from the transition. Subtraction of the lattice contribution
resulted in $\Delta C_{p}(T)$ (inset in Fig. 9). A similar value was obtained
from the DSC measurement (Fig. 10). The change of entropy at the transition
$\Delta S$ has been determined by integration of $\tfrac{\Delta C_{p}(T)}{T}$
over temperature. The values of $\Delta C_{p}$ = 60.5(1) J/(mol K) and $\Delta
S$ = 0.6 J/mol are comparable to transition entropies for $R_{5}$Ir4Si10 and
$R_{2}$Ir3Si5 15, 31. However, they are much larger than obtained for
conventional CDW systems, such as K0.3MoO3 ($\Delta C_{p}$(max) = 8 J/(mol K);
$\Delta S$ = 0.18R)53 and NbSe3 ($\Delta C_{p}$(max) =9 J/(mol K); $\Delta S$
= 0.08R)55. The specific heat anomaly indicates a much sharper transition for
single crystals of Ho2Ir3Si5 than for conventional CDW systems, which is in
agreement with the first-order character of the transition as deduced from
resistivity and magnetic susceptibility data.

##  4 Conclusions

We have established an incommensurately modulated crystal structure of the CDW
phase of Ho2Ir3Si5. The incommensurate modulation is accompanied by a strong
lattice distortion, both of which are important for the modulation of
interatomic distances on zigzag chains of iridium atoms along $\mathbf{c}$.
This is in accordance with the CDW being supported by these zigzag chains.
Similar to the case of Er2Ir3Si5, the rare earth atoms are not directly
involved in the CDW formation. The occurrence of a large lattice distortion
accounts for the sluggish character and large hysteresis of the transition, as
they are apparent in the temperature dependencies of the electrical
resistivity, magnetic susceptibility and specific heat. Another unique feature
of the compounds $R_{2}$Ir3Si5 ($R$ = Lu, Er and Ho) is the extreme
sensitivity of the phase transitions to crystalline order. The present single
crystals of high perfection undergo a CDW transition, while magnetic order is
suppressed down to at least 1.5 K. It is worthwhile to point out that the
previous experiments on polycrystalline material did not observe the CDW
transition, while magnetic order appeared below $T_{N}$ = 5.1 K 32, 33. The
present results on Ho2Ir3Si5 do not confirm the idea that $T_{CDW}$ would
scale with the atomic radius of the rare earth element. While the size of Ho
is equal to that of Er, $T_{CDW}$ is much lower for Ho2Ir3Si5 than for
Er2Ir3Si5. This absence of a correlation in the series $R_{2}$Ir3Si5 is in
contrast to the series of isostructural rare-earth compounds $R_{5}$Ir4Si10
15, where $T_{CDW}$ increases with increasing size of the rare earth element
$R$.

The SXRD data have revealed that the structure of Ho2Ir3Si5 is different from
that of Lu2Ir3Si5 and Er2Ir3Si5. Although the symmetries are the same for all
three compounds, the presence of second-order superlattice reflections in the
CDW phase of Ho2Ir3Si5 has added more complexity to it, requiring modulation
functions with up to second-order harmonics (model A in Section 3.1).

From physical property measurements, a huge hysteresis is found of about 40 K,
with the transition proceeding at 90 K (cooling) and 130 K (warming). The
transition implies severe structural distortions, such that sometimes the
crystal shatters. This is visible in the electrical resistivity, which is
higher after cycling though the transition than before (Fig. 7). The effective
magnetic moment of Ho is found to change at the CDW transition, from 11.38
$\mu_{B}$ to 11.27 $\mu_{B}$ (Fig. 8). This kind of coupling between CDW and
magnetism is rare. It has only been observed in isostructural Er2Ir3Si5. This
feature can probably be attributed to the participation of $4f$ orbitals of Ho
or Er in states near the Fermi level that are involved in CDW ordering. The
fact the effective moment of the rare earth is influenced by the CDW
transition suggests a competition between CDW and magnetic order, with both
interactions presumedly employing the same part of the Fermi surface. In order
to clearly understand the mechanism for this unusual nature one would probably
need microscopic probes of magnetism, such as elastic and inelastic neutron
scattering.

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, using beamline P24. J.-K. Bao acknowledges financial
support from the Alexander-von-Humboldt foundation. The work at TIFR Mumbai
was supported by the Department of Atomic Energy of the Government of India
under Project No. 12-R&D-TFR-5.10-0100. This research has been funded by the
Deutsche Forschungsgemeinschaft (DFG; German Research Foundation)–406658237.

Details of the SXRD data collection, data processing and structural analysis.
This material is available free of charge via the Internet at
http://pubs.acs.org.

## References

  * Grüner 1994 Grüner, G. _Charge Density Waves in Solids_ ; Addison-Wesley: Reading, Massachusetts, 1994
  * Marmeggi et al. 1982 Marmeggi, J. C.; Delapalme, A.; Lander, G. H.; Vettier, C.; Lehner, N. Atomic displacements in the incommensurable charge-density wave in alpha-uranium. _Sol. Stat. Commun._ 1982, _43_ , 577–581
  * Fleming et al. 1981 Fleming, R. M.; DiSalvo, F. J.; Cava, R. J.; Waszczak, J. V. Observation of Charge-Density waves in the cubic spinel structure CuV2S4. _Phys. Rev. B_ 1981, _24_ , 2850–2853
  * Kawaguchi et al. 2012 Kawaguchi, S.; Kubota, Y.; Tsuji, N.; Kim, J.; Kato, K.; Takata, M.; Ishibashi, H. Structural Analysis of Spinel Compound CuV2S4 with Incommensurate Charge-Density Wave. _J. Phys. Conf. Ser._ 2012, _391_ , 012095
  * Okada et al. 2004 Okada, H.; Koyama, K.; Watanabe, K. Two-Step Structural Modulations and Fermi Liquid State in Spinel Compound CuV2S4. _J. Phys. Soc. Jpn_ 2004, _73_ , 3227–3230
  * Ramakrishnan et al. 2019 Ramakrishnan, S.; Schönleber, A.; Hübschle, C. B.; Eisele, C.; Schaller, A. M.; Rekis, T.; Bui, N. H. A.; Feulner, F.; van Smaalen, S.; Bag, B.; Ramakrishnan, S.; Tolkiehn, M.; Paulmann, C. Charge density wave and lock-in transitions of ${\mathrm{CuV}}_{2}{\mathrm{S}}_{4}$. _Phys. Rev. B_ 2019, _99_ , 195140
  * Ślebarski and Goraus 2013 Ślebarski, A.; Goraus, J. Electronic structure and crystallographic properties of skutterudite-related Ce${}_{3}M_{4}$Sn13 and La${}_{3}M_{4}$Sn13 ($M$ = Co, Ru, and Rh). _Phys. Rev. B_ 2013, _88_ , 155122
  * Otomo et al. 2016 Otomo, Y.; Iwasa, K.; Suyama, K.; Tomiyasu, K.; Sagayama, H.; Sagayama, R.; Nakao, H.; Kumai, R.; Murakami, Y. Chiral crystal-structure transformation of $R_{3}$Co4Sn13 ($R$ = La and Ce). _Phys. Rev. B_ 2016, _94_ , 075109
  * Welsch et al. 2019 Welsch, J.; Ramakrishnan, S.; Eisele, C.; van Well, N.; Schönleber, A.; van Smaalen, S.; Matteppanavar, S.; Thamizhavel, A.; Tolkiehn, M.; Paulmann, C.; Ramakrishnan, S. Second-order charge-density-wave transition in single crystals of ${\mathrm{La}}_{3}{\mathrm{Co}}_{4}{\mathrm{Sn}}_{13}$. _Phys. Rev. Materials_ 2019, _3_ , 125003
  * DiMasi et al. 1995 DiMasi, E.; Aronson, M. C.; Mansfield, J. F.; Foran, B.; Lee, S. Chemical pressure and charge-density waves in rare-earth tritellurides. _Phys. Rev. B_ 1995, _52_ , 14516–14525
  * Ru et al. 2008 Ru, N.; Condron, C. L.; Margulis, G. Y.; Shin, K. Y.; Laverock, J.; Dugdale, S. B.; Toney, M. F.; Fisher, I. R. Effect of chemical pressure on the charge density wave transition in rare-earth tritellurides $R$Te3. _Phys. Rev. B_ 2008, _77_ , 035114
  * Banerjee et al. 2013 Banerjee, A.; Feng, Y.; Silevitch, D. M.; Wang, J.; Lang, J. C.; Kuo, H.-H.; Fisher, I. R.; Rosenbaum, T. F. Charge transfer and multiple density waves in the rare earth tellurides. _Phys. Rev. B_ 2013, _87_ , 155131
  * DiMasi et al. 1996 DiMasi, E.; Foran, B.; Aronson, M. C.; Lee, S. Stability of charge-density waves under continuous variation of band filling in LaTe2-xSbx ($0<x<1$). _Phys. Rev. B._ 1996, _54_ , 13587–13596
  * Shim et al. 2004 Shim, J. H.; Kang, J.-S.; Min, B. I. Electronic Structures of $R$Te2 ($R$ = La, Ce): A Clue to the Pressure-Induced Superconductivity in CeTe1.82. _Phys. Rev. Lett._ 2004, _93_ , 156406
  * Ramakrishnan and van Smaalen 2017 Ramakrishnan, S.; van Smaalen, S. Unusual ground states in $R_{5}T_{4}X_{10}$ ($R$ = rare earth; $T$ = Rh, Ir; and $X$ = Si, Ge, Sn): a review. _Rep. Prog. Phys._ 2017, _80_ , 116501
  * Kuo et al. 2020 Kuo, C. N.; Hsu, C. J.; Tseng, C. W.; Chen, W. T.; Lin, S. Y.; Liu, W. Z.; Kuo, Y. K.; Lue, C. S. Charge density wave like behavior with magnetic ordering in orthorhombic Sm2Ru3Ge5. _Phys. Rev. B_ 2020, _101_ , 155140
  * Nakamura et al. 2015 Nakamura, A.; Uejo, T.; Honda, F.; Takeuchi, T.; Harima, H.; Yamamoto, E.; Haga, Y.; Matsubayashi, K.; Uwatoko, Y.; Hedo, M.; Nakama, T.; Onuki, Y. Transport and Magnetic Properties of EuAl4 and EuGa4. _J. Phys. Soc. Jpn_ 2015, _84_ , 124711
  * Shimomura et al. 2019 Shimomura, S.; Murao, H.; Tsutsui, S.; Nakao, H.; Nakamura, A.; Hedo, M.; Nakama, T.; Onuki, Y. Lattice Modulation and Structural Phase Transition in the Antiferromagnet EuAl4. _J. Phys. Soc. Jpn_ 2019, _88_ , 014602
  * Kaneko et al. 2021 Kaneko, K.; Kawasaki, T.; Nakamura, A.; Munakata, K.; Nakao, A.; Hanashima, T.; Kiyanagi, R.; Ohhara, T.; Hedo, M.; Nakama, T.; Onuki, Y. Charge-Density-Wave Order and Multiple Magnetic Transitions in Divalent Europium Compound EuAl4. _J. Phys. Soc. Jpn_ 2021, _90_ , 064704
  * Ramakrishnan et al. 2022 Ramakrishnan, S. et al. Orthorhombic charge density wave on the tetragonal lattice of EuAl4. _IUCrJ_ 2022, _9_ , 378–385
  * Meier et al. 2022 Meier, W. R.; Torres, J. R.; Hermann, R. P.; Zhao, J.; Lavina, B.; Sales, B. C.; May, A. F. Thermodynamic insights into the intricate magnetic phase diagram of EuAl4. _Phys. Rev. B_ 2022, _106_ , 094421
  * Zeng et al. 2022 Zeng, L.; Hu, X.; Wang, N.; Sun, J.; Yang, P.; Boubeche, M.; Luo, S.; He, Y.; Cheng, J.; Yao, D.-X.; Luo, H. Interplay between Charge-Density-Wave, Superconductivity, and Ferromagnetism in CuIr2-xCrxTe4 Chalcogenides. _J. Phys. Chem. Lett._ 2022, _13_ , 2442–2451
  * Roman et al. 2018 Roman, M.; Strychalska-Nowak, J.; Klimczuk, T.; Kolincio, K. K. Extended phase diagram of $R$NiC2 family: Linear scaling of the Peierls temperature. _Phys. Rev. B_ 2018, _97_ , 041103(R)
  * Shimomura et al. 2009 Shimomura, S.; Hayashi, C.; Asaka, G.; Wakabayashi, N.; Mizumaki, M.; Onodera, H. Charge-Density-Wave Destruction and Ferromagnetic Order in SmNiC2. _Phys. Rev. Lett._ 2009, _102_ , 076404
  * Wölfel et al. 2010 Wölfel, A.; Li, L.; Shimomura, S.; Onodera, H.; van Smaalen, S. Commensurate charge-density wave with frustrated interchain coupling in SmNiC2. _Phys. Rev. B_ 2010, _82_ , 054120
  * Shimomura et al. 2016 Shimomura, S.; Hayashi, C.; Hanasaki, N.; Ohnuma, K.; Kobayashi, Y.; Nakao, H.; Mizumaki, M.; Onodera, H. Multiple charge density wave transitions in the antiferromagnets $R$NiC2 ($R$ = Gd, Tb). _Phys. Rev. B_ 2016, _93_ , 165108
  * Kolincio et al. 2017 Kolincio, K. K.; Roman, M.; Winiarski, M. J.; Strychalska-Nowak, J.; Klimczuk, T. Magnetism and charge density waves in $R$NiC2 ($R$ = Ce, Pr, Nd). _Phys. Rev. B_ 2017, _95_ , 235156
  * Maeda et al. 2019 Maeda, H.; Kondo, R.; Nogami, Y. Multiple charge density waves compete in ternary rare-earth nickel carbides, $R$NiC2 ($R$: Y, Dy, Ho, and Er). _Phys. Rev. B_ 2019, _100_ , 104107
  * Kolincio et al. 2020 Kolincio, K. K.; Roman, M.; Klimczuk, T. Enhanced Mobility and Large Linear Nonsaturating Magnetoresistance in the Magnetically Ordered States of TmNiC2. _Phys. Rev. Lett._ 2020, _125_ , 176601
  * Ramakrishnan et al. 2020 Ramakrishnan, S.; Schönleber, A.; Rekis, T.; van Well, N.; Noohinejad, L.; van Smaalen, S.; Tolkiehn, M.; Paulmann, C.; Bag, B.; Thamizhavel, A.; Pal, D.; Ramakrishnan, S. Unusual charge density wave transition and absence of magnetic ordering in Er2Ir3Si5. _Phys. Rev. B_ 2020, _101_ , 060101(R)
  * Sangeetha et al. 2015 Sangeetha, N. S.; Thamizhavel, A.; Tomy, C. V.; Basu, S.; Awasthi, A. M.; Rajak, P.; Bhattacharyya, S.; Ramakrishnan, S.; Pal, D. Multiple charge density wave transitions in single-crystalline Lu2Ir3Si5. _Phys. Rev. B_ 2015, _91_ , 205131
  * Singh et al. 2004 Singh, Y.; Pal, D.; Ramakrishnan, S. Low-temperature studies of the magnetic and superconducting properties of the $R_{2}$Ir3Si5 ($R$ = Y, La, Ce–Nd, Gd–Tm) system. _Phys. Rev. B_ 2004, _70_ , 064403
  * Singh et al. 2005 Singh, Y.; Pal, D.; Ramakrishnan, S.; Awasthi, A. M.; Malik, S. K. Phase transitions in Lu2Ir3Si5. _Phys. Rev. B_ 2005, _71_ , 045109
  * Kuo et al. 2006 Kuo, Y. K.; Sivakumar, K. M.; Su, T. H.; Lue, C. S. Phase transitions in Lu2Ir3Si5: An experimental investigation by transport measurements. _Phys. Rev. B_ 2006, _74_ , 045115
  * Lee et al. 2011 Lee, M. H.; Chen, C. H.; Chu, M.-W.; Lue, C. S.; Kuo, Y. K. Electronically phase-separated charge-density waves in Lu2Ir3Si5. _Phys. Rev. B_ 2011, _83_ , 155121
  * Ramakrishnan et al. 2021 Ramakrishnan, S.; Schönleber, A.; Bao, J.-K.; Rekis, T.; Kotla, S. R.; Schaller, A. M.; van Smaalen, S.; Noohinejad, L.; Tolkiehn, M.; Paulmann, C.; Sangeetha, N. S.; Pal, D.; Thamizhavel, A.; Ramakrishnan, S. Modulated crystal structure of the atypical charge density wave state of single-crystal Lu2Ir3Si5. _Phys. Rev. B_ 2021, _104_ , 054116
  * Clementi et al. 1967 Clementi, E.; Raimondi, D. L.; Reinhardt, W. P. Atomic Screening Constants from SCF Functions. II. Atoms with 37 to 86 Electrons. _J. Chem. Phys._ 1967, _47_ , 1300
  * Meier et al. 2021 Meier, W. R.; Chakoumakos, B. C.; Okamoto, S.; McGuire, M. A.; Hermann, R. P.; Samolyuk, G. D.; Gao, S.; Zhang, Q.; Stone, M. B.; Christianson, A. D.; Sales, B. C. A Catastrophic Charge Density Wave in BaFe2Al9. _Chem. Mater._ 2021, _33_ , 2855–2863
  * Stokes et al. 2011 Stokes, H. T.; Campbell, B. J.; van Smaalen, S. Generation of $(3+d)$-dimensional superspace groups for describing the symmetry of modulated crystalline structures. _Acta Crystallogr. A_ 2011, _67_ , 45–55
  * ho 2 See Supplemental Material at [URL will be inserted by publisher] for details on the diffraction experiments and values of the structural parameters.
  * Schreurs et al. 2010 Schreurs, A. M. M.; Xian, X.; Kroon-Batenburg, L. M. J. EVAL15: a diffraction data integration method based on ab initio predicted profiles. _J. Appl. Crystallogr._ 2010, _43_ , 70–82
  * Sheldrick 2008 Sheldrick, G. M. _SADABS, Version 2008/1_ ; Göttingen: University of Göttingen, 2008
  * van Smaalen 2007 van Smaalen, S. _Incommensurate Crystallography_ ; International Union of Crystallography Monographs on Crystallography; OUP Oxford, 2007
  * Wagner and Schönleber 2009 Wagner, T.; Schönleber, A. A non-mathematical introduction to the superspace description of modulated structures. _Acta Crystallogr. B_ 2009, _65_ , 249–268
  * Petricek et al. 2016 Petricek, V.; Eigner, V.; Dusek, M.; Cejchan, A. Discontinuous modulation functions and their application for analysis of modulated structures with the computing system JANA2006. _Z. Kristallogr._ 2016, _231_ , 301–312
  * Petricek et al. 2014 Petricek, V.; Dusek, M.; Palatinus, L. Crystallographic computing system JANA2006: general features. _Z. Kristallogr._ 2014, _229_ , 345–352
  * Blöchl 1994 Blöchl, P. E. Projector augmented-wave method. _Phys. Rev. B_ 1994, _50_ , 17953–17979
  * Kresse and Furthmüller 1996 Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. _Phys. Rev. B_ 1996, _54_ , 11169–11186
  * Perdew et al. 1996 Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. _Phys. Rev. Lett._ 1996, _77_ , 3865–3868
  * Momma and Izumi 2008 Momma, K.; Izumi, F. VESTA: a three-dimensional visualization system for electronic and structural analysis. _J. Appl. Crystallogr._ 2008, _41_ , 653–658
  * Togo and Tanaka 2015 Togo, A.; Tanaka, I. First principles phonon calculations in materials science. _Scripta Mater._ 2015, _108_ , 1–5
  * Parsons 2003 Parsons, S. Introduction to twinning. _Acta Crystallogr. Section D_ 2003, _59_ , 1995–2003
  * Bartl et al. 1979 Bartl, H.; Schuster, D.; Schroeder, F. A. Verfeinerung der Kristallstruktur der blauen Kalium Molybdaen-bronze, K0.3MoO3, durch Roentgenbeugung. _Zeitschrift fur Kristallographie_ 1979, _149_ , 127–128
  * Kohn 1959 Kohn, W. Image of the Fermi Surface in the Vibration Spectrum of a Metal. _Phys. Rev. Lett._ 1959, _2_ , 393–394
  * Tomić et al. 1981 Tomić, S.; Biljaković, K.; Djurek, D.; Cooper, J.; Monceau, P.; Meerschaut, A. _Solid State Commun._ 1981, _38_ , 109–112
  * Yang et al. 1991 Yang, H. D.; Klavins, P.; Shelton, R. N. Low-temperature physical properties of $R_{5}$Ir4Si10 ($R$ = Dy, Ho, Er, Tm, and Yb) compounds. _Phys. Rev. B_ 1991, _43_ , 7688–7694
  * Ghosh et al. 1993 Ghosh, K.; Ramakrishnan, S.; Chandra, G. Magnetism in the R5Ir4Si10 (R = Ho and Er) systems. _Phys. Rev. B_ 1993, _48_ , 4152–4155