PHYSICAL REVIEW B 112, 035151 (2025)
Tuning incommensurate charge order in Ba
1−x
Sr
x
Al
4
and Ba
1−y
Eu
y
Al
4
Prathum Saraf ,
1
Eleanor M. Clements ,
2
Danila Sokratov ,
1
Shanta Saha,
1
Peter Zavalij ,
3
Thomas W. Heitmann,
4
Jeffrey W. Lynn ,
2,1
Camille Bernal-Choban,
5
Dipanjan Chaudhuri ,
5
Caitlin S. Kengle,
5
Yue Su ,
5
Simon Bettler ,
5
Nathan Manning ,
5
Peter Abbamonte ,
5
Sananda Biswas,
6
Roser Valentí ,
6,7
and Johnpierre Paglione
1,7
1
Maryland Quantum Materials Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
2
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
3
Department of Chemistry, University of Maryland, College Park, Maryland 20742, USA
4
University of Missouri Research Reactor, University of Missouri, Columbia, Missouri 65211, USA
5
Department of Physics and Materials Research Laboratory, The Grainger College of Engineering,
University of Illinois Urbana–Champaign, Urbana, Illinois 61801, USA
6
Institut für Theoretische Physik, Goethe-Universität Frankfurt, 60438 Frankfurt am Main, Germany
7
Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8
(Received 7 May 2025; revised 25 June 2025; accepted 30 June 2025; published 21 July 2025)
The BaAl
4
-type structure family is home to a vast landscape of interesting and exotic properties, with
descendant crystal structures hosting a variety of electronic ground states including magnetic, superconducting,
and strongly correlated electron phenomena. BaAl
4
itself hosts a nontrivial topological band structure, but is
otherwise a paramagnetic metal. However, the other members of the AAl
4
family (A = alkali earth), including
SrAl
4
and EuAl
4
, exhibit symmetry-breaking ground states including charge density wave (CDW) and magnetic
orders. Here we investigate the properties of the solid solution series Ba
1−x
Sr
x
Al
4
and Ba
1−y
Eu
y
Al
4
using
transport, thermodynamic, and scattering experiments to study the evolution of the charge-ordered state as it is
suppressed with Ba substitution to zero near 50% substitution in both systems. Neutron and x-ray diffraction
measurements reveal an incommensurate CDW state in SrAl
4
with c-axis-oriented ordering vector (0, 0,
0.097) that evolves with Ba substitution toward a shorter wavelength. A similar progression is observed in the
Ba
1−y
Eu
y
Al
4
series that also scales with the ordering temperature, revealing a universal correlation between
charge-order transition temperature and ordering vector that points to a critical wave vector that stabilizes CDW
order in both systems. We study the evolution of the phonon band structure in the Ba
1−x
Sr
x
Al
4
system, revealing
that the suppression of the CDW phase matches the suppression of a phonon instability at precisely the same
momentum as observed in experiments, confirming the electron-phonon origin of charge order in this system.
DOI: 10.1103/d8yy-1w3l
I. INTRODUCTION
The evolution of interest in systems harboring charge-
ordered states has produced a plethora of fascinating phenom-
ena and observations that shed light on both cooperative and
competing states in a wide range of materials. Starting with
the discovery and understanding of the role of charge order
in the cuprate phase diagram [1], and more recently with the
varieties of charge order in transition-metal dichalcogenides
[2], time-reversal symmetry breaking at the charge-order tran-
sition in the kagome series AV
3
Sb
5
[3], interplay with strong
correlations in FeGe [4], and nematic order and fluctuations
in BaNi
2
As
2
[5], the interplay of charge, spin, and orbital
degrees of freedom has come into focus as a key question
in quantum materials. Additionally, the effect of charge mod-
ulation and symmetry breaking on topological aspects of
electronic band structures has come to light as a potential
area of interest, for instance in kagome materials [6], and
charge-ordered phases have themselves been proposed to en-
tail nontrivial topologies [7,8].
The family of materials with the BaAl
4
parent struc-
ture type hosts a variety of exotic behaviors [9], including
iron-based superconductivity [10], heavy-fermion physics in
systems such as CeCu
2
Si
2
[11], electronic nematic phases in
BaNi
2
As
2
[5], hidden order in URu
2
Si
2
[12–14], and topolog-
ical spin textures in EuGa
2
Al
2
[15] and EuCd
2
As
2
[16]. The
AAl
4
and AGa
4
series of binary compounds—with A = Ba,
Sr, Ca or Eu—have been known to harbor a variety of charge-
ordered and magnetic ground states [17,18], including charge
density wave (CDW) order at 243 and 140 K in SrAl
4
and
EuAl
4
, respectively, and rare-earth magnetism below ∼20 K
in the Eu-based compounds. However, recent studies of the
BaAl
4
compound itself—a nonmagnetic, metallic material
that does not exhibit any charge-ordered phase—have re-
vealed it to host a crystalline symmetry-protected nontrivial
topology with a three-dimensional Dirac spectrum [19,20]
that likely also exists in the Sr- and Eu-based counterparts
[21]. Charge order in SrAl
4
and EuAl
4
, as well as its anoma-
lous absence in the nearly identical BaAl
4
, have recently
been studied theoretically [22] and experimentally [23–27],
and debate about the origins of CDW order suggest a subtle
sensitivity of the CDW phase to details of the phonon and
electron band structures [22,26].
In this work, we study the evolution of the CDW state in the
Ba
1−x
Sr
x
Al
4
and Ba
1−y
Eu
y
Al
4
solid solution series as a func-
tion of isovalent Sr/Eu substitution (x) in order to verify and
2469-9950/2025/112(3)/035151(9) 035151-1 ©2025 American Physical Society
