# Multiple lattice instabilities and complex ground state in Cs2AgBiBr6

Xing He  Mechanical Engineering and Materials Science, Duke University,
Durham, NC, USA
Matthew Krogstad  Materials Science Division, Argonne National Laboratory,
Lemont, IL, USA
X-ray Sciences Division, Argonne National Laboratory, Lemont, IL, USA
Mayanak K Gupta  Mechanical Engineering and Materials Science, Duke
University, Durham, NC, USA
Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085,
India
Tyson Lanigan-Atkins  Mechanical Engineering and Materials Science, Duke
University, Durham, NC, USA
Chengjie Mao  Mechanical Engineering and Materials Science, Duke University,
Durham, NC, USA
Feng Ye  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, TN, USA
Yaohua Liu  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, TN, USA
Tao Hong  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, TN, USA
Songxue Chi  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, TN, USA
Haotong Wei  Applied Physical Sciences, University of North Carolina at Chapel
Hill, Chapel Hill, NC, USA
Jinsong Huang  Applied Physical Sciences, University of North Carolina at
Chapel Hill, Chapel Hill, NC, USA
Stephan Rosenkranz  Materials Science Division, Argonne National Laboratory,
Lemont, IL, USA
Raymond Osborn  Materials Science Division, Argonne National Laboratory,
Lemont, IL, USA
Olivier Delaire<EMAIL_ADDRESS>Mechanical Engineering and Materials
Science, Duke University, Durham, NC, USA
Department of Physics, Duke University, Durham, NC, USA
Department of Chemistry, Duke University, Durham, NC, USA

###### Abstract

Metal halides perovskites (MHPs) are attracting considerable interest for
optoelectronic applications, with Cs2AgBiBr6 one of the main contenders among
lead-free systems. Cs2AgBiBr6 crystallizes in a nominally double-perovskite
structure, but exhibits a soft lattice with large atomic fluctuations
characteristic of MHPs. While crucial to understand electron-phonon and
phonon-phonon couplings, the spatio-temporal correlations of these
fluctuations remain largely unknown. Here, we reveal these correlations using
comprehensive neutron and x-ray scattering measurements on Cs2AgBiBr6 single-
crystals, complemented with first-principles simulations augmented with
machine-learned neural-network potentials. We report the discovery of an
unexpected complex modulated ground state structure containing several hundred
atoms, arising from a soft-phonon instability of the low-temperature
tetragonal phase. Further, our experiments and simulations both reveal
extensive correlated 2D fluctuations of Br octahedra at finite temperature,
arising from soft anharmonic optic phonons, reflecting very shallow potential
wells. These results provide new insights into the atomic structure and
fluctuations in MHPs, critical to understand and control their thermal and
optoelectronic properties.

## Introduction

Figure 1: Cs2AgBiBr6 crystal structures, Brillouin zones (BZ), and diffuse
scattering. (a,b) Structures of cubic and tetragonal Cs2AgBiBr6. Cs, Ag, Bi
and Br atoms are shown in cyan, silver, purple, and brown respectively. The
tetragonal rotation axis in (b) is the $c$-axis. (c,d) Projection of cubic
structure BZ boundaries in (c) $(L=0)$ plane and (d) $(L=1)$ plane (in double
perovskite conventional notation). Blue lines are the double-perovskite BZ
boundaries, and red lines are the single perovskite BZ boundaries. Green lines
symbolize the location of observed diffuse rods, connecting $\Gamma$ and $X$
points of the double perovskite BZ for $(H,K,L)$ all odd, which corresponds to
the BZ edges for the single perovskite notation. Green diamonds represent $K$
points of double-perovskite BZ, which become superlattice peaks at low $T$.
(e-h) Neutron diffuse scattering data measured in the reciprocal plane $(L=1)$
using CORELLI ($E$-discriminating cross-correlation turned off) at (e) $T=$233
K, (f) 134 K, (g) 100 K and (h) 10 K. The $L$-integration range is 0.96 to
1.04 (r.l.u.). The red rectangle in (e) highlights a diffuse rod. Red and blue
lines in (h) mark the single-perovskite and double-perovskite BZ edges,
respectively. Color maps in (e-h) use a $\log_{10}$ scale. Concentric rings in
neutron data are from diffraction by polycrystalline aluminum and copper parts
of the sample holder (absent from X-ray measurements).

Metal halide perovskites (MHPs) exhibit outstanding performance for
photovoltaic, optoelectronic and radiation detection applications [1, 2, 3, 4,
5], and also attract interest for thermoelectric [6, 7] or thermochromic
applications [8]. The efficiency of MHP solar cells has drastically improved
over the last decade [9, 10], enabled by their excellent carrier transport
properties, including extremely long carrier lifetimes [11, 12, 4, 13, 14,
15]. All-inorganic MHPs present advantages in terms of stability, and enable
the growth of large single-crystals with larger bandgaps well suited for
radiation detection or thermochromic applications [16, 17, 8, 18]. Cs2AgBiBr6
and other lead-free double-perovskite MHPs are attracting increased interest,
in part because of their lower toxicity and improved stability in air compared
to lead-halide systems [19, 20, 21, 22, 23]. Replacing Pb2+ in CsPbBr3 with
Ag+ and Bi3+ lowers the band gap and improves stability [19, 20, 24, 25].
Further, extremely long photocarrier lifetimes were reported [19, 20, 14].

Importantly, computational and experimental investigations of the crystal
structure and optical properties of Cs2AgBiBr6 revealed a clear link between
its structural and optical properties and a strong electron-phonon coupling
[23, 26, 8, 25, 27]. A number of studies in inorganic MHPs also highlighted
the importance of lattice instabilities of the halide octahedra framework, and
of strong phonon anharmonicity on the electron-phonon coupling [28, 29, 30,
31]. Large amplitude atomic fluctuations in MHPs arise from soft and
anharmonic potentials, which also result in very low thermal conductivities
and large thermal expansion coefficients [32, 6, 33, 29, 34, 35, 36, 37].
Cs2AgBiBr6 in particular is predicted to exhibit ultralow lattice thermal
conductivity, which is of fundamental interest and offers promise for
thermoelectric applications [6, 31, 38, 39]. Thus, understanding the atomic
structure and dynamics of Cs2AgBiBr6 is critical to rationalize its
properties. Yet, a detailed understanding of structural dynamics and phonon
anharmonicity in Cs2AgBiBr6 remains elusive owing to a lack of measurements on
high-quality crystals.

Further, several MHPs were recently predicted to exhibit complex ground states
with large unit cells [40], although this prediction remains to be
experimentally confirmed. A number of studies investigated competing low-
energy lattice instabilities, resulting from many shallow local minima in the
potential energy surface [41, 42, 43, 40], and highlighted the coupling
between these instabilities and the electronic states at the edges of the band
gap [44, 41, 45, 46, 40, 47]. Strongly anharmonic phonons and surprising two-
dimensional correlations of Br octahedra tilts were recently revealed in
CsPbBr3 [31], but it remains to be established whether these unusual dynamics
also arise in Pb-free MHPs.

Here, we report a comprehensive investigation of the crystal structure and
lattice dynamics in Cs2AgBiBr6 single-crystals using inelastic neutron
scattering (INS) and diffuse scattering of both neutrons and x-rays. Our
experiments are complemented with first-principles simulations of anharmonic
phonons and structural distortions, extended to large simulation scales with a
surrogate neural-network potential derived from machine-learning. Our
experiments and simulations both reveal a surprisingly complex ground state,
and pervasive 2D correlated fluctuations of Br octahedra tilts.

Figure 2: Diffuse rods in cubic phase and damped low-energy optic branch.
(a,b) Thermal diffuse scattering (TDS) simulations for cubic phase computed
using TDEP 150 K force-constants. Energy integration ranges 0-2 meV and 0-25
meV are shown in (a) and (b) and compared with (d) neutron measurements at 134
K (with CC turned on) and (e) x-ray measurement at 134 K. In all TDS
simulations, the location of Bragg peaks are indicated by yellow crosses.
Panels (c) shows the Br eigenvector weighted phonon dispersions of 150 K cubic
phase. The color maps on dispersions refer to the modulus of Br components in
eigenvectors of each mode. As eigenvectors are normalized to one for each mode
as $\sum_{j}{e_{j}^{2}}=1$, where j is atom and $e_{j}$ is eigenvector, the
minimum of color map is 0 and maximum is 1. (f) $S(\boldsymbol{Q},E)$ along
$(3,K,1)$ calculated from 125 K MLMD trajectory on a 4$\times$20$\times$4
supercell (12800 atoms). The low-energy damped optic branch is responsible to
the diffuse rods observed in neutron and x-ray experiment, as marked as red
segments in (d) and (e). The color map of panel (c) is in linear scale, while
color maps of panel (a-b) and (d-f) are in $\log_{10}$ scale. Concentric rings
in neutron data are from diffraction by polycrystalline aluminum and copper
parts of the sample holder (absent from X-ray measurements).

## Results and Discussion

The average structure of Cs2AgBiBr6 is cubic double-perovskite ($Fm\bar{3}m$)
at room temperature [21, 23, 48]. Compared to single halide perovskites
($ABX_{3}$ prototype, e.g. CsPbBr3), the conventional cell of Cs2AgBiBr6 has
doubled lattice constants and contains 40 atoms, while the primitive cell of
the face centered cubic (FCC) lattice contains 10 atoms. The Ag+ and Bi3+ ions
form an ordered rock-salt sublattice on the $B$ site of the parent single
perovskite. On cooling, Cs2AgBiBr6 undergoes a second-order phase transition
at $T_{c1}\simeq 122$ K to a tetragonal structure with space group $I4/m$,
which was proposed to arise from the freezing of a zone-center soft-mode (out-
of-phase Br octahedra rotations about the $c$ axis, $a^{0}a^{0}c^{-}$ in
Glazer notation) [23, 48, 39]. As shown in Fig. 1(a,b), the resulting
octahedral rotations about the $c$ axis create adjacent layers perpendicular
to this axis having opposite phase. A recent pair distribution function (PDF)
study reported local distortions under high pressure but present even at
ambient condition, although the associated time scale was not investigated
[49].

_Single-crystal diffuse scattering –_ Neutron diffuse scattering measurements
were conducted on a single crystal (mass $\sim 0.2$ g) as a function of
temperature using CORELLI at the Spallation Neutron Source. CORELLI uses a
wide wavelength band to efficiently map large swaths of wave-vectors
$\mathbf{Q}$ in reciprocal space, but also enables the discrimination of
purely elastic vs inelastic signals with a resolution of $\sim 1\,$meV (for
incident neutron energy at 30 meV), through the use of a semi-random chopper
and cross-correlation (CC) technique [50]. Large $\mathbf{Q}$-volumes of
diffuse x-ray scattering (integrating over the phonon energy spectrum) were
also collected with hard x-rays ($E=87.0\,$keV) on Sector 6-ID-D at the
Advanced Photon Source (crystal thickness $\sim 500\,\mu m$). Sharp Bragg
peaks at integer $(H,K,L)$ in both neutron and x-ray experiments indicate the
high quality of the crystals and the absence of extra grains.

Figure 1 illustrates the cubic and tetragonal average crystal structures
(a,b), the correspondence between single and double perovskite Brillouin zones
(BZ) in (c,d), and shows several diffuse scattering maps extracted from
CORELLI data volumes in (e,f,g,h). At all temperatures, weak superstructure
peaks are observed at $(H,K,L)$ all odd (double-perovskite notation),
equivalent to $R$ points of the single-perovskite BZ, corresponding to the
rock-salt ordering of Ag+ and Bi3+ ions on the perovskite $B$-site sublattice.
Strikingly, in the cubic phase, rods of diffuse scattering intensity are
observed in reciprocal planes $L=odd$ in double-perovskite notation, such as
$L=1$ in Fig. 1 (e,f), reminiscent of recent observations in CsPbBr3 [31]. The
rod intensity increases upon cooling in the cubic phase, reaching a maximum
just above $T_{c1}$, as seen in Fig. 1 (e,f) (see also Supplement Fig.S3 and
S4). The rods remain faintly visible at 100 K in the tetragonal phase, see
Fig. 1(g). While they are observed for CORELLI CC either on or off, the
diffuse rods are more intense in the energy-integrated data (CC off) than in
the cross-correlated data (nearly elastic scattering), indicating that the
rods have an inelastic component, although they also extend into the
quasielastic regime, since the cross-correlation energy filtering is
approximately $1$ meV around the elastic channel Fig. 2(d). Recent neutron and
x-ray scattering measurements on CsPbBr3 revealed similar diffuse rods with an
extended quasielastic component over $0\leqslant E\leqslant 2\,$meV, for
transverse-acoustic (TA) modes dominated by Br motions along the edges of the
simple perovskite BZ [31]. First-principles simulations pointed out their
origin in the competition of in-phase vs out-of-phase patterns of collective
PbBr6 octahedra rotations corresponding to the lowest-energy TA phonons at the
$M$ and $R$ points of the simple perovskite BZ.

While the behavior observed in Cs2AgBiBr6 establishes a more general character
in the MHP family of slow correlated fluctuations first observed in CsPbBr3
[31], we also note important differences between the two compounds. In
particular, the intensity along diffuse rods in cubic CsPbBr3 is clearly
modulated with maxima at $M$ and $R$ points [31], whereas the rods in
Cs2AgBiBr6 exhibit no local maximum at $M$ ($X$ point in double perovskite),
and the $R$ points correspond to the Ag/Bi ordering superlattice peaks
($\Gamma$ points in double perovskite). The absence of maxima at $M/X$, can be
related to the different types of transition into the tetragonal phase. The
soft-mode condensation at $\Gamma$ in the double-perovskite is a second-order
transition and yields tetragonal symmetry without introducing superlattice
peaks as the unit cell is unchanged. On the other hand, the $M$ point mode in
CsPbBr3 condenses at a first-order transition and doubles the unit cell. In
addition, the diffuse rods remain strong in the tetragonal phase of CsPbBr3,
but become much weaker in tetragonal Cs2AgBiBr6. Upon cooling Cs2AgBiBr6
deeper in the tetragonal phase, the $X$ point does not condense, but we show
below how a low-lying mode near $K$ points condenses instead (see Fig. 1(h))
and creates a more complex distorted ground state structure than previously
reported [21, 23, 48, 49].

To investigate the phononic origin of the diffuse rods, and anharmonic
renormalization in the cubic phase, we performed ab initio molecular dynamics
(AIMD) simulations at 300 K and 150 K (cubic). The AIMD trajectories were used
as input to the temperature-dependent effective potential (TDEP) method to
obtain renormalized second-order force-constants and associated phonon
dispersions [51, 52, 53]. The renormalized phonon dispersions are shown in
Fig. 2(c) and Supplement Fig. S11. As seen on these plots, the $\Gamma-X$
branch is unstable in harmonic calculations but becomes anharmonically
stabilized by thermal fluctuations, yielding a low-lying flat branch at $E\sim
1.5\,$meV at 300 K, in good agreement with previous simulations [39], and with
our INS measurements of phonon energies shown in Fig. 3. This branch is even
softer at 150 K than at 300 K and results in more intense thermal diffuse
scattering (TDS) (Supplement Fig. S17), consistent with our measurements
showing an increased rod intensity just above $T_{c1}\simeq 122$ K, since the
TDS is proportional to the Bose-Einstein occupation factor and thus highly
sensitive to low-energy phonons. We emphasize that the entire $\Gamma-X$ low-
energy optic branch softens, instead of just the wave-vector ($\Gamma$) of the
soft mode corresponding to the cubic-tetragonal transition. The lowest energy
optic branch along $\Gamma-X$ is dominated by Br octahedra rotations, as
revealed by the phonon dispersions weighted by Br motion amplitudes shown in
Fig. 2 (c) and by phonon animations in the Supplement. The $\Gamma$ mode
involves out-of-phase octahedra rotations like the $R$ mode in the single-
perovskite, while the $X$ mode involves in-phase rotations like the $M$ mode
in the single-perovskite. The dynamical structure factor,
$S(\boldsymbol{Q},E)$, was further computed using the renormalized force-
constants in the cubic phase (see Eq. 1 and details in Appendix). Results at
150 K are shown in Fig. 2(a,b). To compare with CORELLI data, the simulated
$S(\boldsymbol{Q},E)$ was integrated over $0\leqslant E\leqslant 2$ meV
(quasi-elastic channel) to compare with Fig. 2(d), or $0\leqslant E\leqslant
25$ meV (full TDS corresponding to cross-correlation chopper off or to x-ray
diffuse scattering) to compare with Fig. 2(e). The excellent agreement between
the computed TDS and measured diffuse scattering maps, for both neutron and
x-ray scattering, further establishes that the low energy flat branch from Br
motions is responsible for the diffuse rods observed in planes $L=odd$. Due to
the phase factor, these rods have suppressed intensity in planes $L=even$
(supplement Figs. S1 and S2).

While our TDS simulations using renormalized force-constants account for the
observed diffuse scattering signal very well, the TDEP method still describes
the system within an effective second-order harmonic approximation. To fully
include the effect of anharmonicity on the phonon spectral functions, without
the limitations of perturbation theory, we further used machine-learning based
molecular dynamics (MLMD), with a neural-network force-field trained against
AIMD data (see Appendix). Figure 2(f) shows the $S(\boldsymbol{Q},E)$ along
$[3,K,1]$ calculated from the 125 K MLMD trajectory (4$\times$20$\times$4
supercell containing 12800 atoms, see Eq. 4 and details in Appendix). It
clearly reveals the flat low-energy optic branch, which is strongly damped
along the entire $\Gamma-X$ segment and results in a broad spectrum with
quasielastic component $E\leqslant 2$ meV. This behavior is a signature of the
strong anharmonic damping of Br octahedra rotations. The breakdown of these
modes into very low energy fluctuations is responsible for the strong diffuse
rods along $\Gamma-X$, as indicated by the red segment along $[3,K,1]$ marked
in Fig. 2 (d,e). In addition, we performed MLMD on a large
10$\times$10$\times$10 supercell (40000 atoms) to compute the diffuse
scattering (see Appendix). The MLMD reproduced the observed rods from Br
fluctuations, as shown in supplement Figs. S22, S23, and S24, validating the
training of the neural-network force-field.

Figure 3: Inelastic neutron scattering spectra and phonon dispersions. (a,b)
CTAX measurements along at 300 K along (a) $[4,K,0]$ and (b) $[-H,4,-H]$. (c)
HB-3 measurement at $\boldsymbol{Q}=(7,1,1)$ at 300 K. The scattered markers
in (a-c) are measured data and lines are fitted curves using damped harmonic
oscillator models convolving with instrumental resolutions. The inset plot in
(c) shows the scan at $\boldsymbol{Q}=(7,1,1)$ across elastic line (markers).
The fitted Gaussian curve (red lines) centered at energy 0.126 meV. This value
is used to correct the phonon frequency at zone-center. (d) Renormalized
phonon dispersions using cubic phase 300 K AIMD force-constants compared with
experimental data. Markers are fitted phonon energies, and error bars indicate
fitted intrinsic phonon linewidths. (e) $S(\boldsymbol{Q},E)$ along $[4,K,0]$
direction measured at CTAX, generated by stacking scans with different $K$,
comparing with (f) MLMD calculated $S(\boldsymbol{Q},E)$ on a
4$\times$20$\times$4 supercell (12800 atoms). The color maps of panel (e,f)
are in $\log_{10}$ scale.

_Inelastic neutron scattering –_ We now compare our INS measurements with our
phonon simulations and show that low-energy acoustic modes retain a sharp
nature, contrasting strongly with the behavior of soft, broad optic phonons
for Br octahedra rotations. We conducted high-resolution INS on Cs2AgBiBr6
single-crystals using the Cold Neutron Triple-Axis Spectrometer (CTAX) and
HB-3 triple-axis spectrometer at the High Flux Isotope Reactor (details in
Appendix). The TA phonons measured along $[4,K,0]$ and $[-H,4,-H]$ at 300 K
are shown in Fig. 3(a,b). They exhibit clear peaks with relatively narrow
linewidths. In contrast, the spectra at $\boldsymbol{Q}=(7,1,1)$ shown in Fig.
3(c), reveal that the low-energy zone-center optic mode is strongly broadened
at 300 K, with only a broad bump at $\sim 1.5$ meV. Additional CTAX phonon
spectra along $[0,4-K,0]$, $[2+H,2-H,0]$ at 300 K, and at $(3,1,1)$ at several
temperatures are shown in supplement.

The INS spectra were fit with a damped harmonic oscillator (DHO) model
convolved with the experimental resolution (details in supplement). The
resulting phonon energies and linewidths are shown in Fig. 3(d). As can be
seen, our computed renormalized phonon dispersions at 300 K agree very well
with our 300 K INS measurements. In addition, a clear softening of the zone-
center optic mode is observed at 135 K ($E=0.70\,$meV) compared to 300 K
($E=1.44\,$meV), confirming that the zone-center soft-mode condensation in the
cubic phase yields the tetragonal transition [23, 48, 39]. The phonon spectra
measured along $[4,K,0]$ at 300 K are plotted as a color map in Fig. 3(e). The
sharp nature of acoustic modes is captured well by our MLMD
$S(\boldsymbol{Q},E)$ simulations (Fig. 3(f) and supplementary Fig. S21),
which validates our neural-network force-field, and strongly contrasts with
the broad flat optic branch along $\Gamma-X$ shown in Fig. 2 (f).

Figure 4: Lattice instability of tetragonal phase at $K$-points and modulated
ground state. (a) TDS simulation in tetragonal phase (plane $L=1$) integrated
over the energy range 0-25 meV, computed using interpolated harmonic DFT
force-constants. The location of Bragg peaks is indicated by yellow crosses.
(b) Experimental x-ray data (plane $L=1$) measured at 30 K, and zoomed in
portion in (c) comparing data at 30, 50, 100, and 134 K. As seen in (c), $X$
point intensity gradually splits to 4 nearby $K$ points. (d) Phonon
dispersions of harmonic DFT tetragonal phase, with color coding indicating the
modulus of Br eigenvector. Eigenvectors are normalized to unity for each mode
as $\sum_{j}{e_{j}^{2}}=1$, where j is atom and $e_{j}$ is eigenvector, the
minimum of the color map is 0 and maximum is 1. (e) Waterfall plot of x-ray
scattering intensity along $[H,2,3]$ (r.l.u) over a large temperature range.
The $X$ points at $(5,2,3)$, $(6,2,3)$ start to split $\sim 122$ K on cooling,
indicating cubic-tetragonal phase transition. The background discontinuity
$\sim 90$ K is due to the change from Helium to Nitrogen cooling gas, while
the sudden increase of intensity below 40 K indicates the phase transition
from soft-modes near $K$ points condensing. (f) Predicted exaggerated ground
state structure viewed along $c$ and $b$ axis. The color maps of panel (a-c)
are in $\log_{10}$ scale, while (d,e) are in linear scale.

_Tetragonal phase instability and ground state –_ As discussed above, the
tetragonal phase results from a modulation of the cubic structure by out-of-
phase octahedral rotations, due to the soft zone-center optic mode. Indeed,
distorting the cubic structure with the eigenvector of the zone-center mode
resulted in a tetragonal structure with space group $I4/m$, whose parameters
(Table S1) agree well with reported experimental values [23]. Further harmonic
phonon simulations performed on this structure were stable across most of the
BZ, but surprisingly retained a weak instability near the $K$ point,
(0.75,0.75,0) in double perovskite conventional notation, as shown in
supplement Fig. S11 (b). The eigenvector of this unstable mode is illustrated
in supplementary animations. This phonon instability indicates that the true
ground state should exhibit a further distortion and enlarged unit cell.
Indeed, the unstable mode matches well the wave-vectors of the superlattice
peaks observed in the CORELLI measurements at 10 K, see Fig. 1(h) and low-$T$
x-ray data in Fig. 4(b,c) (see note [54]). However, we note that superlattice
peaks appear slightly away from $K$, closer to $(0.8,0.8,0)$. Since the
imaginary phonon frequency of the computed soft-mode at $K$ is small, the
instability is weak and the phase transition occurs at low temperature. As
expected, this instability is sensitive to the exchange-correlation
functionals and the unit cell volume (supplement Fig. S13). Barely unstable
dispersions were obtained by interpolating force-constants (FC) at different
volumes (Fig. 4(d) and Supplement Fig. S14), and subsequently were used to
compute $S(\boldsymbol{Q},E)$, integrated over $0\leqslant E\leqslant 25$ meV
to obtain the TDS in Fig. 4(a). This can directly be compared with the 30 K
x-ray diffuse scattering in Fig. 4(b), showing excellent agreement. We note
that $K$-points are close to the $X$-points in the BZ (Fig. 2 (c,d)) and share
some similarities in their eigenvectors. Both of the low-energy modes at $K$
and $X$ introduce in-phase octahedral rotations. However, within $a-b$ layers,
instead of having a two single-perovskite periodicity, the $K$ soft mode
modulates the rotation with a larger wavelength. As shown in Fig. 4 (f), the
modulation by the soft mode at $K$ results in a large $4\times 4\times 1$
supercell of the original double-perovskite cell, containing 640 atoms
(distortion amplitude exaggerated to help visualize atomic displacements). We
see that in addition to Br displacements mainly in the $a-b$ plane, Cs
displacements occur along the $c$ axis. The modulation resulting from the soft
$K$ mode reduces the symmetry of the tetragonal structure. Our DFT simulations
suggest that the ground state could have space-group $Pm$ (#6), although the
monoclinic distortion angle away from $90^{\circ}$ could be very small. If a
modulation wave vector of $(0.8,0.8,0)$ is used instead, an even larger
$5\times 5\times 1$ supercell (1000 atoms) results, with a possible space
group $Cm$ (#8) (see details in Supplement Section VIII). We note that
distortion energies are exceedingly small (less than 0.1 meV/atom) and thus
sensitive to details of the DFT simulations.

In addition, our low-$T$ x-ray measurements clearly show a broad diffuse
intensity maximum at $X$ points at 134 K (cubic phase) and strengthening at
100 K (tetragonal) and gradually splitting into sharper maxima close to $K$
points, eventually condensing into intense superstructure Bragg spots (Fig.
4(b,c)). A gradual temperature evolution is revealed in Fig. 4(e), which shows
the intensity along $(H,2,3)$ from 300 K to 30 K. Cooling below
$T_{c1}=122\,$K, each $X$-point maximum ($H=\\{5,6\\}$) splits into two maxima
around $K$ points ($H=\\{4.75,5.25\\}$ and $\\{5.75,6.25\\}$ respectively).
Note that the path in Fig. 4(e) is integrated over a wide band to include $K$
points at $(H,2,3\pm 0.25)$. A clear increase in intensity near $K$ occurs
below 40 K, indicating the transition to the distorted ground state. This is
entirely consistent with the condensation of the soft mode near $K$ and its
competition with the $X$ point near-instability (see discussion in Supplement
Sections VI and VII). Using CTAX, we tracked the superlattice peak intensity
around $K=(3.25,0.75,0)$ over a cooling and warming cycle (details in
Supplement Section VII), as shown in the inset of Fig. 4(e). The intensity was
fit to $A\cdot(T_{c2}-T)^{x}+b$, where $A$, $T_{c2}$, $x$ and $b$ are
parameters and $T$ is temperature. The fit results for $T_{c2}$ from cooling
and warming curves are $39.36\pm 0.34$ and $38.78\pm 0.20$ K, respectively.
The coincidence of cooling and warming curves and $T_{c2}$ values supports
that this phase transition is of second order (or weak first order), and
originates from the condensation of a soft mode near the $K$-point.

## Conclusion

In summary, we reported the first momentum-resolved investigations of complex
dynamic fluctuations in the double-perovskite Cs2AgBiBr6 using extensive
neutron and x-ray scattering measurements, rationalized based on first-
principles simulations augmented with machine-learning methods. We found
systematic diffuse scattering rods arising from slow, correlated 2D
fluctuations of Br octahedra derived from damped low-energy optic phonons.
These large fluctuations share similarities with the overdamped zone-boundary
acoustic modes in the single-perovskite CsPbBr3, suggesting a general behavior
of slow correlated 2D fluctuations in MHPs. Critically, a new low-symmetry
phase was discovered at $T\leqslant T_{c2}=39$ K, emerging from a phonon
instability of the tetragonal phase near the $K$-point, and resulting in an
unusually complex ground state with several hundred atoms in the unit cell.
Our results suggest ubiquitous lattice instabilities in the halide perovskite
family resulting from their anharmonic potential energy surface, and reveal a
much more complex ground state than previously known in Cs2AgBiBr6. The
complex anharmonic lattice dynamics yield large thermal fluctuations and
strongly impact the electron-phonon coupling and thus the thermal and
optoelectronic properties of these compounds.

## Materials and Methods

### Synthesis

48 % w/w aq. Hydrobromic acid (HBr) and Bismuth (III) bromide (BiBr3) (metals
basis) with purity of 99 % were purchased from Alfa Aesar, and Silver bromide
(AgBr) (metals basis) with purity of 99 % and Cesium Bromide (CsBr) with 99.9
% purity were purchased from Sigma Aldrich. All the chemicals were directly
used as received without any further purification process. Briefly, Cs2AgBiBr6
was synthesized through a solution-processed method with some modification
[20]. 4.85 g CsBr, 2.15 g AgBr, and 5.11 g BiBr3 were dissolved into 125 mL
HBr solution in a sealed container at 120 $\degree$C on a hot plate after
about two hours stirring. The precursor solution was filtered at 120
$\degree$C before single crystal growth. Finally, large single crystals can be
grown by gradually reducing the heating temperature of hot plate to 80
$\degree$C at a rate of 0.5 $\degree$C/h. To avoid any light induced chemicals
decomposition, the Cs2AgBiBr6 single crystals were grown in the dark
condition.

### X-ray and neutron scattering

#### Diffuse X-ray scattering

Diffuse X-ray scattering data were collected at the Advanced Photon Source on
sector 6-ID-D using an incident energy of 87.0 keV. Sample temperatures from
30 K to 300 K were controlled with He and N2 gas flow. During the
measurements, the samples were continuously rotated about an axis
perpendicular to the beam at 1${}^{\circ}s^{-1}$ over 370∘, with images
recorded every 0.1 s on a Dectris Pilatus 2M detector with a 1-mm-thick CdTe
sensor layer. Three sets of rotation images were collected for each sample at
each temperature to fill in gaps between the detector chips. The resulting
images were stacked into a three-dimensional array, oriented using an
automated peak search algorithm and transformed in reciprocal space
coordinates, allowing S$(\mathbf{Q})$ to be determined over a range of
$\sim\pm$15 $\AA^{-1}$ in all directions. The x-ray scattering intensities are
proportional to S(Q), weighted by the square of the atomic scattering factors
for each element. Further details are given in Ref. [55].

#### Diffuse neutron scattering

Neutron diffuse scattering experiments were conducted using the CORELLI
instrument at SNS, Oak Ridge National Laboratory (ORNL) as a function of
temperature. Unlike conventional single-crystal x-ray or neutron
diffractometers, CORELLI uses a wide wavelength band to cover large swaths of
wave-vectors $\mathbf{Q}$ in reciprocal space, but enables the discrimination
of purely elastic vs inelastic signals with a resolution of $\sim 1\,$meV,
through the use of a semi-random chopper and cross-correlation technique. The
crystal was mounted on an aluminum pin with thin copper wire, with the pin
wrapped in Gd-foil to reduce background, and the assembly was loaded on
closed-cycle refrigerator. Measurements were performed on cooling down to
$T\sim 10\,$K as well as warming to $\sim 400\,$K. At selected temperatures,
the sample was rotated over a wide range of angle ($360^{\circ}$) to collect
large continuous data volumes in reciprocal space and investigate the diffuse
scattering.

#### Inelastic neutron scattering

Inelastic neutron scattering (INS) measurements were conducted on the Cold
Neutron Triple-Axis Spectrometer (CTAX) and HB-3 triple-axis spectrometer at
the High Flux Isotope Reactor (HFIR), ORNL. A few single crystals of
Cs2AgBiBr6 were used with mass 0.2$\sim$0.4ġ. Samples had mosaics of about
30′. Using CTAX, we measured acoustic modes along $[4,K,0]$ at 300 K and 100
K, $[-H,4,-H]$ at 300 K, $[0,4-K,0]$ at 300 K, $[2+H,2-H,0]$ at 300 K, and
optic mode at $\boldsymbol{Q}=(3,1,1)$ at 550 K, 300 K, and 135K. Using HB-3,
we measured the lowest optic mode at $\boldsymbol{Q}=(7,1,1)$ at 300 K.

In CTAX experiments, the fixed final energy mode was used with $E_{f}=5$ meV
for measurement along $[4,K,0]$ at 300 K and 100 K, and with $E_{f}=4.8$ meV
for measurements along $[-H,4,-H]$ at 300 K, $[0,4-K,0]$ at 300 K,
$[2+H,2-H,0]$ at 300 K and $\boldsymbol{Q}=(3,1,1)$ at 550 K, 330 K and 135 K.
PG002 monochromators and analyzers were used. The collimation settings were
30′-$100^{\prime}-80^{\prime}-120^{\prime}$, and energy resolution was $\sim
0.3$ meV at zero energy transfer. The high-order energy contamination of
neutron beam was removed by a cooled Be filter. Samples were mounted in the
$(H,K,0)$ plane using Bragg peaks $(4,0,0)$ and $(0,4,0)$ for measurements
along $[4,K,0]$ and $[2+H,2-H,0]$. The scattering plane $(3H,K,H)$ defined by
$(3,-1,1)$ and $(3,1,1)$ was used for measurements at
$\boldsymbol{Q}=(3,1,1)$, along $[-H,4,-H]$, and $[0,4-K,0]$. While
$[-H,4,-H]$ is not within the $(3H,K,H)$ scattering plane, the required tilts
were reachable with the instrument goniometer. We note that $[4,K,0]$ and
$[0,4-K,0]$ both correspond to wave vectors along $\langle 001\rangle$, and
are TA modes and longitudinal acoustic (LA) modes from $\Gamma$ to $X$,
respectively. Along $[2+H,2-H,0]$ and $[-H,4,-H$, we probe the two non-
degenerate TA modes from $\Gamma$ to $K$, which have distinct polarizations.

In the HB-3 experiment, the fixed final energy mode was used with $E_{f}=13.5$
meV. PG002 monochromators and analyzers were used. The collimation setting was
30′-$40^{\prime}-40^{\prime}-70^{\prime}$, and the energy resolution was $\sim
1$ meV at zero energy transfer. The sample was mounted in the $(H,K,0)$ plane.
$\boldsymbol{Q}=(7,1,1)$ was reached with tilts of the instrument goniometer.

### First-Principles Simulations

First-principles phonon simulations were performed using VASP [56, 57, 58, 59]
in the cubic and tetragonal phases of Cs2AgBiBr6, with a special attention
paid to anharmonic effects. We first describe the results of phonon
simulations within the small-displacement approach and the harmonic
approximation, as implemented in Phonopy [60], in the cubic phase
($Fm\overline{3}m$, space group 225), using a $2\times 2\times 2$ supercell of
the double-perovskite cell with experimental lattice constant 11.2695 $\AA$
from ref [61] (details in Supplement). These harmonic simulations result in
unstable phonon dispersions, as shown in Supplement Fig. S8 and Fig. S9. The
unstable mode at the $\Gamma$ point is triple-degenerate, corresponding to
out-of-phase rotations of Br octahedra about the three $\langle
100\rangle_{\rm cubic}$ axes. Modulating the cubic structure according to the
eigenvector of one of these unstable modes results in the tetragonal
structure. The structure was then relaxed using VASP until the force on each
atom was smaller than $10^{-4}$ eV$/\AA$, yielding the expected $I4/m$
symmetry (space group number 87), see details in Supplement. The relaxed
structure has lattice constants $a=b=11.162\,\AA$, $c=11.488\,\AA$, which
compare well with experimental values $a=7.8794\times\sqrt{2}=11.143\ \AA$,
$c=11.324\,\AA$ [23]. After relaxing the tetragonal structure, further
harmonic phonon simulations were performed. The phonon dispersions were then
nearly stable, with only a weak instability around the $K$ point
((0.75,0.75,0) in double perovskite conventional notation) as shown in
Supplement Fig. S8. This instability indicates that the true ground state
exhibits a further modulated distortion and enlarged unit cell. Indeed, the
location of the instability around $K$ matches well the wave-vectors of the
superlattice peaks observed in the Corelli measurements at 10 K. Since the
imaginary phonon frequency around this $K$ point mode is small, the
instability is weak and the phase transition happens at low temperature, below
$40$ K.

The dynamical structure factor $S(\mathbf{Q},E)$ was computed for the cubic
and tetragonal phases using either the harmonic or renormalized force-
constants to obtain phonon frequencies $\omega_{s}$ and eigenvectors $e_{ds}$.
The simulated phonon intensity was calculated as [62]:

$\displaystyle S({\bf Q},E)\propto$
$\displaystyle\sum\limits_{s,\mathbf{\tau}}\frac{1}{\omega_{s}}\left|\sum\limits_{d}\frac{\overline{b_{d}}}{\sqrt{M_{d}}}\rm{exp}(-W_{d})\rm{exp}(i\mathbf{Q}\cdot\mathbf{d})(\mathbf{Q}\cdot\mathbf{e}_{ds})\right|^{2}$
$\displaystyle\times\langle
n_{s}+1\rangle\delta(\omega-\omega_{s})\delta(\mathbf{Q}-\mathbf{q}-\mathbf{\tau})$
(1)

where $\overline{b_{d}}$ is neutron scattering length of atom $d$,
$\boldsymbol{Q}=\boldsymbol{k}-\boldsymbol{k^{\prime}}$ is the wave vector
transfer, and $\boldsymbol{k^{\prime}}$ and $\boldsymbol{k}$ are the final and
incident wave vector of the scattered particle, q the phonon wave vector,
$\omega_{s}$ the eigenvalue of the phonon corresponding to the branch index
$s$, $\bm{\tau}$ is the reciprocal lattice vector, $d$ the atom index in the
unit cell, $\exp(-2W_{d})$ the corresponding Debye-Waller factor, and
$n_{s}=\left[\exp\left(\frac{\hbar\omega_{s}}{k_{\rm
B}T}\right)-1\right]^{-1}$ is the mean Bose-Einstein occupation factor. The
phonon eigenvalues and eigenvectors in Eq. 1 were obtained by solving
dynamical matrix using Phonopy [60]. The thermal diffuse scattering (TDS)
signals are obtained by integrating $S(\boldsymbol{Q},E)$ over energy.

### Machine-Learning augmented Molecular Dynamics (MLMD)

We have extended the molecular dynamics simulation to a larger scale to access
the well-resolved dynamics in ($\boldsymbol{Q}$, $E$) space using a machine-
learned force-field (MLFF) based on a neural-network potential. To generate a
force-field that describes the dynamics over a broad temperature range, we
generated AIMD training datasets over a range of temperatures using VASP [56,
57, 58, 59]. These simulations were performed with the strongly constrained
and appropriately normalized (SCAN) exchange-correlation functional [63]. We
performed AIMD on a 40-atom unit cell, with a plane wave kinetic energy cutoff
of 400 eV and a 2$\times$2$\times$2 Monkhorst-Pack grid of $k$-points [64].
The self-consistent convergence threshold for electronic minimization was set
to $10^{-6}$ eV. We ran simulations from 100 K to 900 K in 100 K intervals,
with each trajectory lasting about 3-5 ps and a time step of 2 fs. All AIMD
simulations used the 300 K experimental lattice constant 11.27 $\AA$ [23]. The
temperature of the system was controlled by a Nosé-Hoover thermostat. We then
used the DEEPMD [65] code to generate a machine-learned force-field (MLFF)
based on a neural network, which reproduces as best as possible the AIMD
dynamics. The generated MLFF was used to compute the pair distribution
function and mean squared displacements and checked against AIMD results. The
LAMMPS package was then used for MLMD classical simulations with the trained
MLFF [66]. We calculated the $S(\mathbf{Q},E)$ from MLMD trajectories at 125 K
and 300 K along $[4,K,0]$ and $[3,K,1]$. For these simulations, we used large
supercells (4$\times$20$\times$4 supercell containing 12800 atoms), and
computed the atomic trajectories up to $\sim 100$ ps (with a time step of 1
fs) within an NVT ensemble. The simulation at 125 K and 300 K used
experimental lattice constants 11.23 and 11.27 $\AA$ respectively [23]. These
extensive simulations provide momentum and energy resolutions of $\sim
0.1\,\mathrm{\AA^{-1}}$ and $\sim$0.1 meV, respectively, enabling us to
resolve the dispersions. In neutron scattering experiments, the scattering
intensity has contributions from both the coherent dynamical structure factor
($S_{\mathrm{{coh}}}^{IJ}$($\boldsymbol{Q}$, $E$)) attributed to correlated
dynamics between $I^{\mathrm{{th}}}$ and $J^{\mathrm{{th}}}$ element, and
incoherent dynamical structure factor
($S_{\mathrm{{inc}}}^{I}$($\boldsymbol{Q}$, $E$)) from uncorrelated self
dynamics of the $I^{\mathrm{{th}}}$ element, respectively defined as:

$S_{\mathrm{coh}}^{IJ}(\boldsymbol{Q},E)=\frac{1}{2\pi}\int_{0}^{\infty}\sum_{i,j}e^{-i\boldsymbol{Q}\cdot(R_{i}^{I}(0)-R_{j}^{J}(t))}e^{iEt/\hslash}dt$
(2)
$S_{\mathrm{inc}}^{I}(\boldsymbol{Q},E)=\frac{1}{2\pi}\int_{0}^{\infty}\sum_{i}e^{-i\boldsymbol{Q}\cdot(R_{i}^{I}(0)-R_{i}^{I}(t))}e^{iEt/\hslash}dt$
(3)

where $R_{i}^{I}$($t$) is the $i^{th}$ atom position of element type $I$ at
time $t$, and the summation indices $i$ and $j$ represent the sum over
$I^{th}$ and $J^{th}$ kind of elements in the simulation cell. The total
measured neutron scattering intensity is then proportional to:

$S_{\mathrm{tot}}(\boldsymbol{Q},E)\propto\sum_{I,J}b_{\mathrm{coh}}^{I}b_{\mathrm{coh}}^{J}\,S_{\mathrm{coh}}^{IJ}(\boldsymbol{Q},E)+\sum_{J}(b_{\mathrm{inc}}^{J})^{2}\,S_{\mathrm{inc}}^{J}(\boldsymbol{Q},E)$
(4)

where $b_{\mathrm{coh}}^{I}$ and $b_{\mathrm{inc}}^{I}$ are the coherent and
incoherent scattering length of $I^{\mathrm{th}}$ element and summation index
$I$ and $J$ runs over different elements in the unit cell.

Diffuse scattering intensity calculations were performed with MLMD using a
10$\times$10$\times$10 supercell containing 40000 atoms. We ran MLMD
simulations at 125 K and 300 K with durations of about 100 ps with 1 fs time
steps. We computed the total scattering intensity, $S$($\boldsymbol{Q}$) at
0.1 ps interval ($\sim$1000 configurations) according to:

$S(\boldsymbol{Q})\propto\int_{0}^{\infty}\lvert\sum_{I}b_{\mathrm{coh}}^{I}\sum_{i\in
I}e^{-i\boldsymbol{Q}\cdot R_{i}(t)}\rvert^{2}dt$ (5)

where $i$ is an index running over all atoms of type $I$, and
$b_{\mathrm{coh}}^{I}$ is the coherent neutron scattering length of atom type
$I$. The diffuse scattering intensity of pair $I-I$ is defined as:

$S^{I-I}(\boldsymbol{Q})\propto(b_{\mathrm{coh}}^{I})^{2}\int_{0}^{\infty}\lvert\sum_{i\in
I}e^{-i\boldsymbol{Q}\cdot R_{i}(t)}\rvert^{2}dt$ (6)

where index $i$ enumerates atoms of element type $I$. Note
$S(\boldsymbol{Q})\neq\sum\limits_{I}S^{I-I}(\boldsymbol{Q})$.

The $S$($\boldsymbol{Q}$) and $S^{I-I}(\boldsymbol{Q})$ computed at different
time points were then averaged to obtain the diffuse scattering pattern.

## Acknowledgements

We thank Jacky Even for interesting discussions. We thank Olle Hellman for
providing access to the TDEP software package. XH, MKG and OD were supported
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division, under Award No. DE-SC0019978. TLA
was supported by the U.S. Department of Energy, Office of Science, Basic
Energy Sciences, Materials Sciences and Engineering Division, under Award No.
DE-SC0019299. Work at the Materials Science Division at Argonne National
Laboratory (characterization, x-ray and neutron scattering measurements and
analysis) was supported by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences, Materials Sciences and Engineering Division.
The synthesis of materials is based upon work supported by the Department of
Energy / National Nuclear Security Administration under Award Number(s) DE-
NA0003921. This research used resources at the High Flux Isotope Reactor and
Spallation Neutron Source, both DOE Office of Science User Facilities operated
by the Oak Ridge National Laboratory. This research used resources of the
Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science
User Facility, operated for the DOE Office of Science by Argonne National
Laboratory under Contract No. DE-AC02-06CH11357. Theoretical calculations were
performed using resources of the National Energy Research Scientific Computing
Center, a U. S. DOE Office of Science User Facility supported by the Office of
Science of the U. S. Department of Energy under Contract No. DE-
AC02-05CH11231.

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