# Transverse structural modulation in nematic SrAl4 and elucidation of its
origin in the BaAl4 family of compounds

Sitaram Ramakrishnan<EMAIL_ADDRESS>Department of Quantum Matter, AdSE,
Hiroshima University, Higashi-Hiroshima 739-8530, Japan  Surya Rohith Kotla
Laboratory of Crystallography, University of Bayreuth, 95447 Bayreuth, Germany
Hanqi Pi  Beijing National Laboratory for Condensed Matter Physics, Institute
of Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physics, University of Chinese Academy of Sciences, Beijing 100049, China
Bishal Baran Maity  Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Mumbai 400005, India  Jia
Chen<EMAIL_ADDRESS>Zhejiang Laboratory, Hangzhou 311121, China
Jin-Ke Bao  Department of Physics, Materials Genome Institute and
International Center for Quantum and Molecular Structures, Shanghai
University, Shanghai 200444, People’s Republic of China  Zhaopeng Guo  Beijing
National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  School of Physics,
University of Chinese Academy of Sciences, Beijing 100049, China  Masaki Kado
Department of Quantum Matter, AdSE, Hiroshima University, Higashi-Hiroshima
739-8530, Japan  Harshit Agarwal  Laboratory of Crystallography, University of
Bayreuth, 95447 Bayreuth, Germany  Claudio Eisele  Laboratory of
Crystallography, University of Bayreuth, 95447 Bayreuth, Germany  Minoru
Nohara  Department of Quantum Matter, AdSE, Hiroshima University, Higashi-
Hiroshima 739-8530, Japan  Leila Noohinejad  P24, PETRA III, Deutsches
Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany  Hongming
Weng<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
School of Physics, University of Chinese Academy of Sciences, Beijing 100049,
China  Srinivasan Ramakrishnan  Department of Physics, Indian Institute of
Science Education and Research, Pune, 411008, India  Arumugam Thamizhavel
Department of Condensed Matter Physics and Materials Science, Tata Institute
of Fundamental Research, Mumbai 400005, India  Sander van Smaalen  smash@uni-
bayreuth.de Laboratory of Crystallography, University of Bayreuth, 95447
Bayreuth, Germany

###### Abstract

At ambient conditions SrAl4 adopts the BaAl4 structure type with space group
$I4/mmm$. It undergoes a charge-density-wave (CDW) transition at $T_{CDW}$ =
243 K, followed by a structural transition at $T_{S}$ = 87 K. Temperature-
dependent single-crystal X-ray diffraction (SXRD) leads to the observation of
incommensurate superlattice reflections at $\mathbf{q}=\sigma\,\mathbf{c}^{*}$
with $\sigma=0.1116$ at 200 K. The CDW has orthorhombic symmetry with the
superspace group $Fmmm(0\,0\,\sigma)s00$, where $Fmmm$ is a subgroup of
$I4/mmm$ of index 2. Atomic displacements represent a transverse wave, and
they are mainly along one of the diagonal directions of the $I$-centered unit
cell. The breaking of fourfold rotational symmetry is indicative of the
presence of nematic order in the material. The orthorhombic phase realized in
SrAl4 is analogous to that found in EuAl4, albeit with the presence of higher
order satellite reflections (up to $m=3$) and a shorter modulation wave
vector. A possible non-trivial band topology has prevented the determination
by density functional theory (DFT) of the mechanism of CDW formation. However,
DFT reveals that Al atoms dominate the density of states near the Fermi level,
thus, corroborating the SXRD measurements. SrAl4 remains incommensurately
modulated at the structural transition, where the symmetry lowers from
orthorhombic to $\mathbf{b}$-unique monoclinic. The present work draws a
comparison on the modulated structures of non-magnetic SrAl4 and magnetic
EuAl4 elucidating their similarities and differences, and firmly establishing
that although substitution of Eu to Sr plays little to no role in the
structure, the transition temperatures are affected by the atomic sizes. We
have identified a simple criterion, that correlates the presence of a phase
transition with the interatomic distances. Only those compounds $X$Al4-xGax
($X$ = Ba, Eu, Sr, Ca; $0<x<4$) undergo phase transitions, for which the ratio
$c/a$ falls within the narrow range $2.51<c/a<2.54$.

##  I Introduction

The manifestation of charge density waves (CDWs) was initially found to occur
in quasi-one-dimensional (1D) electronic systems, like NbSe3 and K0.3MoO3 [1,
2, 3]. These compounds possess Fermi surfaces with co-planar sections that
allow so-called Fermi surface nesting (FSN), thus explaining the stabilization
of CDWs. The nesting vector of the periodic structure becomes the wave vector
of the CDW in the valence bands, as well as the wave vector of the modulation
of the atomic positions (periodic lattice distortion). Recent research has
expanded the criteria for the occurrence of CDWs [4]. Materials need not
support 1D or quasi-two-dimensional (2D) electron bands, but they can involve
complex three-dimensional (3D) electronic system, if certain conditions are
fulfilled. The mechanism of stabilization of CDW in these systems is provided
by $\mathbf{q}$-dependent electron-phonon coupling (EPC). Several examples
exist of 3D compounds with CDWs, which include CuV2S4 [5, 6, 7, 8],
$R$3Co4Sn13 ($R$ = La, Ce) [9, 10, 11], $R$5Ir4Si10 ($R$ = Dy, Ho, Er, Yb, Lu)
[12], Sm2Ru3Ge5 [13, 14], TmNiC2 [15, 16], $R_{2}$Ir3Si5 ($R$ = Lu, Er, Ho)
[17, 18, 19], CuIr2-xCrxTe4 [20], and kagome materials like $A$V3Sb5 ($A$ = K,
Rb, Cs) [21, 22, 23, 24, 25, 26] and FeGe [27, 28, 29].

Compounds $X$Al4 / $X$Ga4 ($X$ = Eu, Ca, Sr, Ba) crystallize in the tetragonal
BaAl4 structure type with space group $I4/mmm$. These systems have attracted a
lot of attention in the recent years, because of their properties as
topological quantum materials [30, 31, 32, 33, 34, 35]. EuAl4, containing
divalent Eu, is a 3D CDW compound that also undergoes four successive
antiferromagnetic (AFM) transitions below 20 K [36, 37, 38, 39, 40]. It has
been reported to have a chiral spin structure and skyrmions [40, 41], similar
to skyrmions reported in other divalent Eu-based systems, such as EuPtSi [42,
43]. A skyrmion state of magnetic order, and the non-trivial band topology
were furthermore established by measurements of the topological Hall
resistivity, the muon-spin rotation and relaxation ($\mu$SR), and the
magnetostriction [32, 44, 34]. Our recent investigation into the modulated
structure of EuAl4 reveals a breaking of the fourfold rotational symmetry,
resulting in an orthorhombic $Fmmm$ symmetry of the CDW state below 145 K.
This finding suggests that the material is a strong candidate for being
nematic [39] much like other CDW systems such as Ba(Ni,Co)2(As,P)2 [45, 46,
47, 48] and CsV3Sb5 [49, 50]. There is no evidence of a lattice distortion.
The reduction in symmetry is brought about solely by the transverse CDW
modulation in EuAl4.

Replacement of the divalent rare-earth element Eu by divalent alkaline earth
metals Ba, Sr and Ca results in isostructural compounds. SrAl4 and CaAl4
undergo CDW transitions or other structural transitions (STs). However, BaAl4
does not undergo any phase transition [51, 33]. Out of all the materials in
the $X$Al4 series, the compound that is most similar to EuAl4 would be SrAl4.
SrAl4 exhibits a CDW transition at $T_{CDW}=243$ K, which is at a much higher
temperature than the CDW transition of EuAl4 [52, 53]. Isostructural CaAl4
undergoes a ST at 443 K, which is at an even higher temperature than for SrAl4
[54]. SrAl4 undergoes a second transition at $T_{S}$ = 87 K [52]. It was
suggested that the symmetry of the low-temperature phase is monoclinic [52].
The mechanism of CDW formation in either SrAl4 or EuAl4 presently is not
understood, as both compounds possess a complex 3D electronic structure [36,
55, 52].

Here, we present the results of temperature-dependent single-crystal x-ray
diffraction (SXRD) experiments on SrAl4. We find that the CDW transition is
accompanied by a reduction in symmetry from tetragonal to orthorhombic,
entirely analogous to EuAl4 [39], albeit with larger amplitudes for the
modulation in SrAl4. The network of Al atoms governs the CDW in SrAl4, which
is similar to the situation of EuAl4 [39]. This finding is supported by solid
solution samples of isostructural SrAl4-xSix and SrAl4-xGex, since the CDW is
suppressed by the disorder at the Al site, and induces superconductivity in
the case of Si-doping [56, 57]. From SXRD and physical properties
measurements, we confirm that below 100 K SrAl4 undergoes a second phase
transition that is characterized by a monoclinic lattice distortion, but
across which the CDW modulation remains incommensurate. In this paper, we
present the modulated crystal structure of SrAl4, and we discuss the
similarities and subtle differences in the CDWs of SrAl4 and EuAl4.
Furthermore, we establish that the value of $c/a$ must be within the narrow
range $2.51<c/a<2.54$, in order for the compound to undergo a phase
transition. This allows us to predict whether new materials developed in this
series could or could not undergo a CDW or structural transition.

##  II Experimental and computational details

###  II.1 Crystal growth

Single crystals of SrAl4 were grown by the Al self-flux method according to
[52]. Crystals were grown at three places, using similar methods.

At the Lab. of Crystallography in Bayreuth, the elements strontium (Alfa
Aesar, 99.95% purity) and aluminium (Alfa Aesar, 99.9995%) were filled into an
alumina crucible in the ratio 1:19. The crucible was sealed in an evacuated
quartz-glass ampoule. It was heated to a temperature of 1173 K and held at
this temperature for 2 hours. The crucible was then cooled to 823 K with a
rate of 0.5 K/hour, at which point the crystals were separated from the molten
metal by centrifugation. Small crystals were selected and annealed in vacuum
for 72 hours at 723 K. The 1:4 stoichiometry of the product was confirmed by
structure refinement against SXRD data. The resulting crystals A were used for
the SXRD experiments.

At the Tata Institute in Mumbai, the elements were filled into an alumina
crucible with the ratio 1:23. Crystal growth was at 1323 K for 24 hours, after
which it was cooled to 973 K with a rate of 1 K/hour, followed by
centrifugation. The longer growth time resulted in crystal B that is larger
than crystal A. This crystal was employed for the susceptibility measurements
without annealing. At the Department of Quantum Matter in Hiroshima, the same
procedure was followed, resulting in crystal C that was used for the
measurement of the electrical resistivity.

###  II.2 Single-crystal x-ray diffraction data collection

SXRD experiments were performed at Beamline P24 of PETRA III at DESY in
Hamburg, employing a four-circle Huber diffractomter with Euler geometry, and
radiation of wavelength 0.50000 Å. The temperature of the crystal was
controlled by a CRYOCOOL open-flow helium cryostat. Complete data sets were
measured at temperatures of 293, 200, 120, 100, 75 and 20 K, covering all
phases. Each run of data collection comprises 3640 frames, corresponding to a
rotation of the crystal over 364 deg, which was repeated 10 times. These data
were binned to a data set of 364 frames of 1 deg rotation and 10 s exposure
time [58]. Further details are provided in the Supplemental Material [59].

###  II.3 Single-crystal x-ray diffraction data processing

The EVAL15 software suite [60] was used for processing the SXRD data. Each
temperature comprises two runs, one with and another without 2$\theta$ offset
of the detector. The two runs were integrated separately, and subsequently
merged in the module ANY of EVAL15. SADABS [61] was used for scaling and
absorption correction with Laue symmetry $4/mmm$ for the data measured at 293
K (periodic phase), with Laue symmetry $mmm$ for the runs collected at 200,
120 and 100 K (CDW phase), and with Laue symmetry $2/m$ ($\mathbf{b}$-unique)
at 20 K (low-temperature phase). The resulting reflection files were imported
into the software JANA2006 [62, 63] for structure refinements. Tables 1 and 2,
and Tables S2–S4 in the Supplemental Material [59] give details of the
analysis, crystallographic information, atomic coordinates and modulation
amplitudes.

Table 1: Crystallographic data of crystal A of SrAl4 at 293 K (periodic phase) and 200 K (CDW phase). Temperature (K) | 293 | 200
---|---|---
Crystal system | Tetragonal | Orthorhombic
Space/Superspace group |  $I$4/$mmm$ | $Fmmm(00\sigma)s00$
No. [64] | 139 | 69.1.17.2
$a$ (Å) | 4.4893(2) | 6.3326(4)
$b$ (Å) | 4.4893 | 6.3331(5)
$c$ (Å) | 11.2764(5) | 11.2541(5)
Volume (Å3) | 227.26(3) | 451.35(4)
Wavevector q | - |  0.1116(2)$\mathbf{c}^{*}$
$Z$ | 2 | 4
Wavelength (Å) | 0.50000 | 0.50000
Detector distance (mm) | 260 | 260
$2\theta$-offset (deg) | 0, 25 | 0, 25
$\chi$-offset (deg) | -60 | -60
Rotation per image (deg) | 1 | 1
$(\sin(\theta)/\lambda)_{max}$ (Å-1) | 0.746821 | 0.745874
Absorption, $\mu$ (mm-1) | 4.844 | 4.878
Tmin, Tmax | 0.3118, 0.3522 | 0.3121, 0.3515
Criterion of observability | $I>3\sigma(I)$ | $I>3\sigma(I)$
No. of reflections measured, |  |
$(m=0)$ | 630 | 493
$(m=1)$ | - | 969
$(m=2)$ | - | 971
Unique reflections (obs/all), |  |
$(m=0)$ | 137/139 | 197/211
$(m=1)$ | - | 323/392
$(m=2)$ | - | 94/403
$R_{int}$ $(m=0)$ (obs/all) | 0.0241/0.0241 | 0.0229/0.0229
$R_{int}$ $(m=1)$ (obs/all) | - | 0.0979/0.0978
$R_{int}$ $(m=2)$ (obs/all) | - | 0.0845/0.1010
No. of parameters | 9 | 25
$R_{F}$ $(m=0)$ (obs) | 0.0144 | 0.0276
$R_{F}$ $(m=1)$ (obs) | - | 0.0452
$R_{F}$ $(m=2)$ (obs) | - | 0.1463
$wR_{F}$ $(m=0)$ (all) | 0.0186 | 0.0306
$wR_{F}$ $(m=1)$ (all) | - | 0.0612
$wR_{F}$ $(m=2)$ (all) | - | 0.1767
$wR_{F}$ all (all) | 0.0186 | 0.0494
GoF (obs/all) | 1.28/1.27 | 1.69/1.35
$\Delta\rho_{min}$, $\Delta\rho_{max}$(e Å-3) | -0.29, 0.47 | -1.89, 3.89
Table 2: Amplitudes of the modulation functions of crystal A at 200 K for superspace group $Fmmm(0\,0\,\sigma)s00$. The relative coordinates, $x$, $y$ and $z$ of the basic position are also specified. Values of modulation amplitudes have been multiplied by the corresponding lattice parameter, in order to obtain values in Å.  Atom | Sr | Al1 | Al2
---|---|---|---
$x$ | 0 | 0.25 | 0
$y$ | 0 | 0.25 | 0
$z$ | 0 | 0.25 | 0.38360(12)
$A_{1,x}\,a$ (Å) | 0.2323(9) | 0.2274(25) | 0.2325(24)
$A_{1,y}\,b$ (Å) | 0 | 0 | 0
$A_{1,z}\,c$ (Å) | 0 | 0 | 0
$B_{1,x}\,a$ (Å) | 0 | 0 | 0.0661(18)
$B_{1,y}\,b$ (Å) | 0 | 0.0731(24) | 0
$B_{1,z}\,c$ (Å) | 0 | 0 | 0
$A_{2,x}\,a$ (Å) | 0 | 0 | 0
$A_{2,y}\,b$ (Å) | 0 | 0 | 0
$A_{2,z}\,c$ (Å) | 0.0084(15) | 0.0058(40) | 0.0015(36)
$B_{2,x}\,a$ (Å) | 0 | 0 | 0
$B_{2,y}\,b$ (Å) | 0 | 0 | 0
$B_{2,z}\,c$ (Å) | 0 | 0 | -0.0003(37)
$A_{3,x}\,a$ (Å) | 0.0230(18) | 0.0218(51) | 0.0137(45)
$A_{3,y}\,b$ (Å) | 0 | 0 | 0
$A_{3,z}\,c$ (Å) | 0 | 0 | 0
$B_{3,x}\,a$ (Å) | 0 | 0 | 0.0161(44)
$B_{3,y}\,b$ (Å) | 0 | -0.0112(86) | 0
$B_{3,z}\,c$ (Å) | 0 | 0 | 0

###  II.4 Density Functional Theory (DFT) Calculations

Density functional theory (DFT) calculations were performed for the tetragonal
crystal structure of SrAl4 with space group of $I4/mmm$, employing the Vienna
ab initio simulation package (VASP) [65]. The Projector-augmented-wave (PAW)
method [66, 67] and generalized gradient approximations (GGA) [68] with
Perdew-Burke-Ernzerhorf (PBE) type pseudopotentials were chosen to deal with
exchange-correlations.

The conventional unit cell was fully relaxed with a $12\times 12\times 4$
$k$-mesh sampling until the energy convergence tolerance fell below $10^{-7}$
eV and the force was less than $10^{-2}$ eV/Å. The cutoff energy of the plane-
wave basis was chosen as $520$ eV. The DFT optimized results are close to the
experimental values (Table 1). An $18\times 18\times 18$ $k$-mesh was
implemented for the Brillouin zone (BZ) integral sampling of the primitive
unit cell. The Fermi surface was calculated by using the WannierTools package
[69] with tight-binding Hamiltonian constructed by the WANNIER90 code [70,
71], which is based on the maximally localized Wannier function (MLWF) method
[72]. The bare charge susceptibility is carried out using a $k$-mesh of
$200\times 200\times 200$. The phonon spectrum calculations were performed
within the framework of density functional perturbation theory (DFPT) [73, 74]
and the finite displacement method, implemented by the PHONOPY [75] package
combined with VASP, and by Quantum ESPRESSO [76, 77].

###  II.5 Physical properties

The dc electrical resistivity was measured between 6 and 296 K, employing a
standard four-probe method. The small size of the specimen prevented the
identification of lattice directions, so that the single experimental run has
resulted in $\rho(T)$ along an unspecified, general direction. A commercial
superconducting quantum interference device (SQUID) magnetometer (MPMS5,
Quantum Design, USA) was used to measure dc magnetic susceptibility in a field
of 0.1 T as a function of temperature from 2 to 300 K. A commercial physical
property measurement system (PPMS, Quantum Design, USA) was used for measuring
the specific heat data. For all transport and bulk measurements, data was
collected during both warming and cooling runs, in order to investigate the
character of the transitions.

##  III Results and discussion

###  III.1 The orthorhombic CDW phase

SrAl4 has been reported to undergo two phase transitions: one at $T_{CDW}=243$
K and another below $90$ K [52]. Cooling the crystal from 293 to 200 K
revealed superlattice reflections with $\mathbf{q}=0.1116(2)\,\mathbf{c}^{*}$
at 200 K, similar to the diffraction of isostructural EuAl4 below $145$ K [37,
39]. The data measured at $200$ K (incommensurate CDW phase) could be refined
successfully (Table 1). Like for EuAl4 [37, 39], we do not observe any lattice
distortion within the CDW phase, thus preserving the tetragonal lattice
symmetry for the basic structure.

In order to elucidate the modulated CDW structure at $200$ K, we have tested
different superspace groups as symmetry of the crystal structure (See Tables
S2 and S3 in the Supplemental Material [59]). The three candidate models are
based on symmetries $I4/mmm$ (model A) and its maximal orthorhombic subgroups
$Immm$ (model B) and $Fmmm$ (model C). The latter symmetry resulted in the
best fit to the SXRD data (Section S2 in the Supplemental Material [59]). The
analysis thus gave model C with superspace symmetry $Fmmm(0\,0\,\sigma)s00$ as
structure model for the incommensurately modulated CDW state of SrAl4.

Although the modulated crystal structure of SrAl4 below 243 K is isostructural
to that of EuAl4 below 145 K, there are some differences. Unlike EuAl4, we
have observed higher order superlattice reflections in the SXRD of SrAl4 (Fig.
1). Consequently, the modulation functions include higher-order harmonic
amplitudes.

Figure 1: Excerpts of the reciprocal layers of diffraction for (a)–(d)
$(0\,k\,l)$ and (e)–(f) $(h\,0\,l)$, as reconstructed from the measured SXRD
data [78]. (a)–(d) and (e)–(f) display scattering for temperatures of 293,
200, 100 and 20 K, as indicated. Indices are given for selected main
reflections. Selected satellite reflections are indicated by arrows. Up to
$3^{rd}$-order satellites are visible at 20 and 100 K. Dark gaps are due to
insensitive pixels between the active modules of the PILATUS3 X CdTe 1M
detector.

Structure refinements indicate a significantly better fit to the satellite
reflections for model CII with up to $3^{rd}$-order harmonics than for model
CI with up to $2^{nd}$-order harmonics (Table 3).

Table 3: Fit to the SXRD data at 200 K for different models with superspace group $Fmmm(00\sigma)s00$. Model CI incorporates up to $2^{nd}$-order harmonics, and model CII incorporates up to $3^{rd}$-order harmonics. Criterion of observability: $I>3\sigma(I)$. Model | CI | CII
---|---|---
$R_{F}(m=0)$(obs/all) (%) | 2.76/2.80 | 2.76/2.80
$R_{F}(m=1)$(obs/all) (%) | 5.04/5.17 | 4.52/4.65
$R_{F}(m=2)$(obs/all) (%) | 18.70/25.42 | 14.63/19.69
No. of unique reflections |  |
$m=0$ (obs/all) | 197/211 | 197/211
$m=1$ (obs/all) | 323/392 | 323/392
$m=2$ (obs/all) | 94/403 | 94/403
No. of parameters | 20 | 25

Therefore, we selected model CII with up to $3^{rd}$-order harmonics in the
modulation functions for further analysis (Tables 1 and 2, and Table S4 in
[59]).

Upon further cooling below 200 K, the length of the modulation wave vector
gradually shrinks [Fig. 2(d)].

Figure 2: Lattice parameters and modulation wave vector within the temperature
range 20–293 K. (a) Lattice parameters ($F$-centered setting) relative to
their values at $T=293$ K, with $a(293)=6.3488(3)$, $b(293)=6.3488$ and
$c(293)=11.2764(3)4$ Å. (b) The lattice parameter $\beta$ in deg. (c) Volume
of the unit cell. (d) Component $\sigma$ of $\mathbf{q}=(0,\,0,\,\sigma)$,
which remains incommensurate down to 20 K.

As a consequence, main reflections and nearby satellite reflections are not
always resolved at $120$ K and below (compare the streaky maxima in Fig. 1). A
reliable extraction was not possible of the values for the integrated
intensities of individual Bragg reflections for $T<120$ K.

###  III.2 Location of the CDW

SrAl4 is isostructural to EuAl4 with a tetragonal crystal structure at room
temperature [79], that is preserved as basic structure at lower temperatures
(Fig. 3).

Figure 3: Crystal structure of SrAl4 with space group $I4/mmm$ in the periodic
phase at 293 K. Depicted is the $I$-centered unit cell with basis vectors
$\mathbf{a}_{I}$, $\mathbf{b}_{I}$ and $\mathbf{c}_{I}$. Orange spheres
correspond to the Sr atoms; green-bluish spheres represent Al1 atoms; and pink
spheres stand for Al2 atoms. Shortest interatomic distances are: $d$[Sr–Sr] =
4.4893(4) Å, $d$[Al1–Al1] = 3.1774(2) Å, $d$[Al2–Al1] = 2.7034(6) Å and
$d$[Al2–Al2] = 2.6250(15) Å. The crystal structure was drawn using VESTA [80].

The modulation amplitudes exhibit the following features in comparison to
those of EuAl4. Firstly, the largest modulation has an amplitude of 0.23 Å
along $\mathbf{a}$ for all atoms, which is about twice the value of 0.12 Å for
the atoms of EuAl4 (Table 2 and [39]). In addition, SrAl4 features
$3^{rd}$-order harmonic amplitudes of values of about 10% of the
$1^{st}$-order harmonic coefficients.

Secondly, exclusively the atom Al1 has a modulation along $\mathbf{b}$ of
amplitude 0.073 Å, which is only 17% larger than in EuAl4. Again, this
modulation has a $3^{rd}$-order harmonic contribution of about 15% of the
$1^{st}$-order harmonic, which is in antiphase with the $1^{st}$-order
harmonic and thus responsible for the obviously anharmonic shape of the
Al1–Al1 distance along the incommensurate coordinate $t$ (Fig. 4).

Figure 4: $t$-Plot of interatomic distances (Å) $d$[Al1–Al1], $d$[Al2–Al1] and
$d$[Al2–Al2] at 200 K, where the first atom is the central atom. The number on
each curve is the number of the symmetry operator that is applied to the
second atom of the bond pair. Symmetry operators are listed in Table S6 in the
supporting information.

The $2^{nd}$-order harmonic amplitudes attain values close to zero.

The shortest Al1–Al1 contacts are the only short intermetallic distances with
a significant variation with the phase of the modulation. This is furthermore
illustrated by Table 4, where the modulations of interatomic distances are
given for both EuAl4 and SrAl4. Apart from a general expansion of the latter,
variations of distances due to the modulations are nearly equal in the two
compounds. Therefore, in analogy to EuAl4, we conclude that the Al1 atoms
govern the formation of the CDW.

Table 4: Selected structural information on the CDW phases of EuAl4 [39] and
SrAl4 (present work). The Crystal structures have superspace group
$Fmmm(0\,0\,\sigma)s00$.

Compound |  EuAl4 |  SrAl4
---|---|---
$T_{CDW}$ (K) | 145 | 243
$T$ (K) | 70 | 200
$a$ (Å) | 6.1992(4) | 6.3326(4)
$b$ (Å) | 6.2001(4) | 6.3331(5)
$c$ (Å) | 11.1477(3) | 11.2541(5)
$V$ (Å3) | 428.47(4) | 451.34(6)
$\sigma$ | 0.1781(3) | 0.1116(2)
Distance Al1–Al1a,b along $\mathbf{a}$
max (Å) | 3.102(3) | 3.169(11)
min (Å) | 3.100(3) | 3.166(11)
avg (Å) | 3.101(2) | 3.168(5)
Distance Al1–Al1c,d along $\mathbf{b}$
max (Å) | 3.224(3) | 3.293(16)
min (Å) | 2.976(3) | 3.040(16)
avg (Å) | 3.101(2) | 3.168(8)
Distance Al2–Al1e,f,g,h
max (Å) | 2.669(2) | 2.728(8)
min (Å) | 2.651(2) | 2.672(8)
avg (Å) | 2.659(2) | 2.698(8)
Distance Al2–Al2i
max (Å) | 2.567(3) | 2.624(6)
min (Å) | 2.566(2) | 2.619(6)
avg (Å) | 2.567(2) | 2.621(6)

Given are the maximum (max), minimum (min) and average (avg) interatomic
distances, with the symmetry operation applied to the second atom.
a,b Symmetry codes for Al1–Al1: $(-x+1,-y+1/2,-z+1/2)$, $(-x+1/2,-y,-z+1/2)$
c,d Symmetry codes for Al1–Al1: $(-x+1/2,-y+1,-z+1/2)$, $(x,y+1/2,z-1/2)$
e,f,g,h Symmetry codes for Al2–Al1: $(x,y,z)$, $(x-1/2,y-1/2,z)$,
$(-x,-y+1/2,-z+1/2)$, $(-x+1/2,-y,-z+1/2)$
i Symmetry code for Al2–Al2: $(-x,y,-z+1)$

###  III.3 Criteria for the formation of CDW in BaAl4 type compounds

Only a few of the isostructural compounds $X$Al4-xGax ($X$ = Ba, Eu, Sr, Ca;
$0<x<4$) undergo phase transitions. We make the observation, that structural
and CDW phase transitions take place in compounds that have the ratio $c/a$
within a narrow range: $2.51<c/a<2.54$ (Fig. 5).

Figure 5: Value of $T_{c}$ = $T_{S}$ or $T_{CDW}$ as a function of the ratio
$c/a$ for eleven compounds $X$Al4-xGax. Only four compounds undergo phase
transitions.

EuGa4 has a CDW phase transition for pressures exceeding 2 GPa [36]. One might
then speculate, that the ratio $c/a$ of EuGa4 at high pressures will be within
the critical range (Fig. 5). This criterion is useful to predict CDWs in new
compounds of this family. At present there is no information regarding whether
SrAl2Ga2 exhibits CDW or other structural transitions.

###  III.4 Evidence of monoclinic distortion below 100 K

The orthorhombic phase is realized through the orthorhombic symmetry of the
CDW modulation displacements (Table 2), while the lattice and basic structure
remain tetragonal (Fig. 2 and Table S4). The structural transition at
$T_{S}=87$ K is accompanied by a lattice distortion. In particular, we find
the angle $\beta$ to be different from 90 deg within the low-temperature phase
[Fig. 2(b)]. These values indicate that the low-temperature phase is
$\mathbf{b}$-unique monoclinic, in agreement with Nakamura et al. [52].

The structural phase transition leads to a twinned crystal, such that the
monoclinic lattice distortion is visible as split reflections in $q$-scans
along the direction of $\mathbf{c}^{*}$. For example, for the reflection
$(0\,2\,4)$, a secondary peak appears below $100$ K between the main
reflection and first-order satellite (Fig. S1 in the Supplemental Material
[59]). Furthermore, these $q$-scans show that the incommensurate CDW satellite
reflections persist into the low-temperature phase. Across $T_{S}$, they
continue to grow in intensity and the length of $\mathbf{q}$ continues to
decrease [Fig. 2(d)].

Due to overlap between main and satellite reflections, as it is the result of
the short modulation wave vector [Fig. 2(d)], structure refinements failed.
However, measured intensities are of sufficient quality to distinguish between
tetragonal, orthorhombic and monoclinic symmetries on the basis of $R_{int}$
values for averaging equivalent reflections. These quantities clearly favor
monoclinic, $\mathbf{b}$-unique symmetry for the low-temperature phase (Table
5).

Table 5: Comparison of models on basis of SSG and $R_{int}$ at 20 K. Criterion of observability: $I>3\sigma(I)$ SSG | $I4/mmm(00\sigma)0000$ | $Fmmm(00\sigma)s00$ |  $F$2/$m(\sigma{{}_{1}}0\sigma{{}_{2}})00$ |
---|---|---|---|---
$R_{int}$ ($m=0$)(obs/all)$\%$ | 5.07/5.07 | 3.78/3.79 | 1.62/1.62 |
$R_{int}$ ($m=1$)(obs/all)$\%$ | 38.47/38.47 | 30.86/30.86 | 5.33/5.35 |
$R_{int}$ ($m=2$)(obs/all)$\%$ | 64.42/64.43 | 38.41/38.41 | 6.23/9.16 |
Unique ($m=0$) (obs/all) | 46/50 | 138/202 | 255/344 |
Unique ($m=1$) (obs/all) | 75/106 | 317/381 | 567/732 |
Unique ($m=2$) (obs/all) | 24/108 | 204/405 | 296/741 |

###  III.5 Electronic Band Structure and Phonons

We have calculated the electronic structure and topological properties of
SrAl4 for its $I4/mmm$ crystal structure. Dispersion relations along high-
symmetry directions within the Brillouin zone are given in Fig. 6(a,b).

Figure 6: The bulk band structures of the tetragonal crystal structure
$I4/mmm$ of SrAl4. (a) Electronic band structure along high-symmetry
directions, as obtained without SOC. (b) Electronic band structure calculated
with SOC. The dashed lines indicate the Fermi energy, $E_{F}$. All band
crossings are gapped out by SOC, except the crossing of two bands along the
M-$\Gamma$ direction, labeled LD6 (green) and LD7 (orange). (c) The density-
of-states (DOS) of the primitive unit cell with SOC. The green line gives the
total DOS; the red line gives the contributions of $2s$ and $2p$ orbitals of
Al; and the blue line stands for contributions of the $3d$ orbital of Sr $d$.
(d) Phonon dispersion relations obatined by DFPT calculations for a $3\times
3\times 2$ supercell of the conventional unit cell. (e) First Brillouin zone
of the primitive unit cell. (f) Hole pockets, and (g) electron pockets of the
Fermi surface.

In the absence of SOC, several band crossings exist between the highest
valence band and the lowest conduction band near the Fermi level [Fig. 6(a)].
When the SOC is considered, most band crossings are gapped, except one
crossing along the line M-$\Gamma$ [Fig. 6(b)]. Unfortunately, it appears at
around $E=0.2$ eV above the Fermi level, which is not easy to find
experimentally.

Using the software IRVSP [81], we found that these two bands belong to the
irreducible representations of point group $4mm$, marked LD6 (green line) and
LD7(orange line), respectively [Fig. 6(b)]. This indicates that the band
crossing is a topologically protected Dirac point. It follows that SrAl4 is a
Dirac nodal line semimetal without SOC and becomes a Dirac semimetal when the
SOC is considered, like EuAl4 [36, 39].

The total and atom-projected density of states (DOS) near $E_{F}$ are given in
Fig. 6(c). They reveal that Al states (red line) dominate the DOS near the
Fermi level. This observation parallels the situation observed for EuAl4 [39].
Consequently, it is plausible to infer that the CDW is associated with the Al
atoms rather than the Sr atoms.

Additional information on the mechanism of CDW formation in SrAl4 might be
obtained from the phonon dispersion relations and the Fermi surface. Figure
6(d) shows the phonon dispersion relations, as computed with aid of a $3\times
3\times 3$ supercell of the primitive basic cell, containing 135 atoms.
Density functional perturbative theory (DFPT) [73, 74] was used with a
$4\times 4\times 2$ $k$-mesh. No imaginary frequencies or soft modes can be
identified, indicating that electron-phonon coupling (EPC) is too weak to
induce a CDW transition. This result is further checked by the application of
alternative computational methods. For the same $3\times 3\times 3$ supercell,
phonon dispersion relations were computed by the finite displacement method.
Small imaginary frequencies were obtained on the line M-$\Gamma$, when a
$3\times 3\times 3$ $k$-mesh was used for the computation (see Fig. S9(a) in
the Supplementary Material [59]). These imaginary frequencies are eliminated,
when the number of $k$-points is increased toward to $5\times 5\times 5$ (Fig.
S9(b) in [59]). The software Quantum Espresso [76, 77] leads to the same
conclusion [compare Figs. S9(b),(c) and Fig. 6(d)].

The Fermi surface of SrAl4 comprises hole pockets centered on the point M
[Fig. 6(f)], as well as electron pockets surrounding $\Gamma$ and those
centered on point P [Fig. 6(g)]. To further check whether Fermi surface
nesting (FSN) could responsible for this CDW, the bare charge susceptibility
[82] was calculated, employing a $k$-mesh of $200\times 200\times 200$, and
considering four conduction bands and four valence bands near the Fermi level,

$\displaystyle\lim_{\omega\rightarrow
0}\chi_{0}^{\prime\prime}(\boldsymbol{q},\omega)/\omega$
$\displaystyle=\sum_{n,n^{\prime},\boldsymbol{k}}\delta\left(\varepsilon_{n,\boldsymbol{k}}-\varepsilon_{F}\right)\delta\left(\varepsilon_{n^{\prime},\boldsymbol{k}+\boldsymbol{q}}-\varepsilon_{F}\right)$
(1) $\displaystyle\chi_{0}^{\prime}(\boldsymbol{q})$
$\displaystyle=\sum_{n,n^{\prime},\boldsymbol{k}}\frac{f\left(\varepsilon_{n,\boldsymbol{k}}\right)-f\left(\varepsilon_{n^{\prime},\boldsymbol{k}+\boldsymbol{q}}\right)}{\varepsilon_{n,\boldsymbol{k}}-\varepsilon_{n^{\prime},\boldsymbol{k}+\boldsymbol{q}}},$

where $f\left(\varepsilon_{n,\boldsymbol{k}}\right)$ is the Fermi-Dirac
distribution function. The real and imaginary components of the bare charge
susceptibility of the SrAl4 on the $k_{z}$ = $0.11$ $c^{*}$ plane are depicted
as a function of wave vector $\mathbf{q}$ in Fig. S10(a,c) in the Supplemental
Material [59]. Both components exhibit peak values at the center. We further
conducted additional calculations of the bare charge susceptibility along the
$k_{z}$ line with $k_{x}$ = $k_{y}$ =$0$, and present the outcomes in Fig.
S10(b,d). The real component shows a peak near the experimentally observed
$\mathbf{q}$ vector, whereas the imaginary part does not. This discrepancy
indicates that Fermi surface nesting (FSN) is insufficient to induce the
charge density wave (CDW) in SrAl4. These conclusions are consistent with the
analysis previously done for SrAl4 and EuAl4 [36, 52, 39], concluding that a
clear mechanism of the CDW cannot be determined directly.

###  III.6 Electrical resistivity

The temperature dependence of the electrical resistivity was measured on a
small single crystal of SrAl4 (crystal C). The upward turn of the resistivity
upon cooling of the crystal is characteristic for a CDW transition (Fig. 7).

Figure 7: Temperature dependence of the electrical resistivity $\rho(T)$ of
SrAl4. Phase transitions are reflected by an anomaly at 243 K and hysteresis
about 90 K.

A second anomaly with hysteresis is observed about $T_{S}=90$ K The observed
features are similar to those reported in [52].

###  III.7 Magnetic susceptibility

Figure 8: Temperature dependence of the magnetic susceptibility $\chi$ for
along $\mathbf{a}$ and $\mathbf{c}$ directions in a field of 0.1 T. The CDW
transition is not clearly visible unlike the low temperature ST much like what
has been reported in [52]. A hysteresis observed at $T_{s}$ indicates that the
transition is of first-order.

The magnetic susceptibility $\chi$ was measured in both directions along
$\mathbf{a}$ and $\mathbf{c}$ as shown in Figure 8. Unlike resistivity, it is
hard to detect an anomaly in the CDW phase at $243$ K from the $\chi$ data,
much like the previous works done on the same material and also for
isostructural EuAl4 [52, 39]. The second ST occurs at a much lower temperature
at around $60$ K rather than $90$ K, probably due to the fast thermal ramp
rate employed during the measurement. A clear hysteresis is observed at $60$ K
suggesting the ST towards monoclinic symmetry from orthorhombic is of first-
order. The rise we see below $15$ K is due to the inevitable presence of trace
amounts of paramagnetic impurities in the starting elements used in making the
crystal.

###  III.8 Specific heat

The temperature dependence of the specific heat ($C_{p}$) shows anomalies
about $T_{CDW}$ (Fig. 9), which are more pronounced than observed for EuAl4
[39].

Figure 9: Temperature dependence of the specific heat $C_{p}$ of SrAl4 for
3–280 K and for both heating and cooling. Major fluctuations are observed for
the transition near $T_{CDW}=243$ K. No anomaly could be detected near the
second structure transition at $T_{S}=87$ K.

The measured $C_{p}(T)$ data do not show any anomaly at the temperature
$T_{S}$ of the structural phase transition (Fig. 9). This might be related to
the effects of annealing, which removes lattice defects, as previously
described for CuV2S4 [8]. The absence of an obvious anomaly in $C_{p}(T)$ at
$T_{S}$ then might be the result of a suppressed transition, as it is caused
by the abundance of lattice defects in the as-grown crystal used for the
$C_{p}$ measurement.

The presence of lattice defects in as-grown material might also be responsible
for the sluggish appearance of the anomaly in the $C_{p}(T)$ data at the CDW
transition (Fig. 9).

##  IV Conclusions

From SXRD experiments and the physical properties measurements we confirm that
SrAl4 undergoes two phase transitions. At $T_{CDW}=243$ K a CDW transition
takes place, at which the crystal symmetry is lowered from tetragonal to
orthorhombic. For the structural transition at $T_{S}=87$ K, SXRD
unambiguously shows that the low-temperature phase is $\mathbf{b}$-axis
monoclinic, and that the CDW is virtually unaffected by this transition.

Replacement of non-magnetic Sr from magnetic Eu has a minimal role in the
crystal structure, but does affect the value of $T_{CDW}$, which can be
explained by different atomic radii of Sr and Eu. This idea has been expanded
towards CaAl4 and BaAl4. More generally, isostructural compounds with the
tetragonal BaAl4 structure type exist for $X$Al4-xGax ($X$ = Ba, Eu, Sr, Ca;
$0<x<4$) We could show that only those compounds undergo phase transitions for
which $c/a$ falls within the narrow range $2.51<c/a<2.54$ (Fig. 5).

Both SXRD and electronic band structure calculations support the
interpretation that the network of Al atoms is the key contribution to CDW
formation, much like in EuAl4. The system possibly possesses a non-trivial
band topology and a complex Fermi surface, preventing the mechanism of CDW
formation to be simply uncovered in SrAl4. The breaking of the fourfold
rotational symmetry through the transverse character of the CDW modulation
suggests that SrAl4 is a nematic CDW.

We surmise that it may be possible to realise multiple phases in EuAl4, as the
latter is proposed to go towards monoclinic symmetry in the AFM phases [39].
For CaAl4 there may exist an intermediary orthorhombic phase between its
tetragonal and monoclinic phases.

###### Acknowledgements.

Single crystals of SrAl4 were grown by Kerstin Küspert at the Laboratory of
Crystallography in Bayreuth, and by Ruta Kulkarni at the Tata Institute of
Fundamental Research in Mumbai. We thank Yilin Wang, Chao Cao, Simin Nie and
Zhijun Wang for useful discussions about theoretical calculations and
exploration of CDW mechanisms. We thank M. Tolkiehn and C. Paulmann for their
assistance at Beamline P24. 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. Beamtime was allocated for proposal I-20210805. J. Chen is
supported by China Postdoctoral Science Foundation (No. 2022M722916). This
work was partly supported by JSPS KAKENHI Grant Numbers JP21K03448 and
JP23H04630. The research at the University of Bayreuth has been funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–406658237.

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