Assessment of dynamic structural instabilities across 24 cubic inorganic
halide perovskites
Ruo Xi Yang,
1, 2
Jonathan M. Skelton,
3
Estelina L. da Silva,
4
Jarvist M. Frost,
5
and Aron Walsh
1, 6, a)
1)
Department of Materials, Imperial College London, Exhibition Road, London SW7 2AZ,
UK
2)
Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd, Berkeley, CA 94720,
USA
3)
Department of Chemistry, University of Manchester, Manchester M13 9PL,
UK
4)
Instituto de Dise˜no para la Fabricaci´on y Producci´on Automatizada, MALTA Consolider Team,
Universitat Polit`ecnica de Val`encia, Val`encia, Spain, 46022 Valencia, Spain
5)
Department of Physics, Imperial College London, Exhibition Road, London SW7 2AZ,
UK
6)
Department of Materials Science and Engineering, Yonsei University, Seoul 120-749,
Korea
(Dated: 20 November 2020)
Metal halide perovskites are promising candidates for next-generation photovoltaic and optoelectronic ap-
plications. The flexible nature of the octahedral network introduces complexity when understanding their
physical behavior. It has been shown that these materials are prone to decomposition, phase competition, and
the local crystal structure often deviates from the average space group symmetry. To make stable phase-pure
perovskites, understanding their structure–composition relations is of central importance. We demonstrate,
from lattice dynamics calculations, that the 24 inorganic perovskites ABX
3
(A = Cs, Rb; B = Ge, Sn, Pb; X =
F, Cl, Br, I) exhibit instabilities in their cubic phase. These instabilities include cation displacements, octahe-
dral tilting, and Jahn-Teller distortions. The magnitudes of the instabilities vary depending on the chemical
identity and ionic radii of the composition.The tilting instabilities are energetically dominant, and reduce
as the tolerance factor increases, whereas cation displacements and Jahn-Teller type distortions depend on
the interactions between the constituent ions. We further considered representative tetragonal, orthorhombic
and monoclinic perovskites phases to obtain phonon-stable phases for each composition. This work provides
insights into the thermodynamic driving force of the instabilities and will help guide synthesis in material
screening.
I. INTRODUCTION
Since the discovery of photoconductivity in the cae-
sium lead halides (CsPbX
3
[X = Cl, Br, I])
1
, the semicon-
ducting properties of halide perovskites have attracted
significant research attention, including analogous com-
pounds based on Sn and Ge.
2–4
Interest has since ex-
panded to the hybrid organic-inorganic perovskites with
potential applications including field-effect transistors,
5
photovoltaics,
6,7
and light-emitting diodes.
8
This fam-
ily of materials display a unique combination of physical
and chemical properties, including fast ion and electron
transport,
9–11
long carrier diffusion lengths,
12
and high
quantum efficiency.
13
The crystallography of lead halide perovskites dates
back to the 1950s, where the high-temperature crystal
structures of the CsPbX
3
series were determined to be
the archetypal cubic perovskite structure (space group
P m
¯
3m). The same structure was later reported for the
CH
3
NH
3
PbX
3
series.
14
In all cases, phase transitions to
lower symmetry perovskite phases are observed as the
temperature is reduced, e.g. in CsPbCl
3
there is a tran-
a)
Electronic mail: a.walsh@imperial.ac.uk
sition to a tetragonal phase at 320 K, an othorhombic
phase at 316 K, and a monoclinic phase at 310 K.
15
The nature of the high temperature cubic phase of the
halide perovskites has received less attention. Analysis
of the X-ray pair distribution functions of CH
3
NH
3
SnBr
3
suggested that the the local cubic symmetry was broken
with significant distortions of the corner-sharing octa-
hedral network.
16
It was recently confirmed, from both
inelastic X-ray and neutron total scattering that the cu-
bic phase of CH
3
NH
3
PbI
3
is also symmetry broken.
17–19
These observations have been associated with rotational
disorder of the molecular CH
3
NH
3
+
cation.
Relatively little work has been reported on the inor-
ganic halide perovskite counterparts. It was shown by X-
ray total scattering that CsPbX
3
nanocrystals always ex-
hibit orthorhombic tilting of the octahedra within locally-
ordered subdomains.
20
Furthermore, low-frequency Ra-
man scattering has confirmed that CsPbBr
3
undergoes
dynamical polar fluctuations at 300 K, even though
analysis of XRD measurement suggests the structure
to be cubic at that temperature.
21
Evidence of sym-
metry breaking has also been reported from analysis of
the temperature-dependant photoluminescence of Cs and
CH
3
NH
3
compounds.
22
In this work, we demonstrate that vibrational instabil-
ities are common to inorganic halide perovskites in the
2
ABX
3
(A = Cs, Rb; B = Ge, Sn, Pb; X = F, Cl, Br, I)
family. The associated dynamic disorder is a consequence
of the flexibility associated with the corner-sharing net-
work of inorganic octahedra, which includes rigid-unit
tilting modes, distortions of the octahedra, and cation
displacements. Through first-principles lattice-dynamics
calculations, we assess the chemical and thermodynamic
driving forces for these instabilities and discuss the con-
sequences for the material properties.
II. CLASSIFICATION OF STRUCTURAL
INSTABILITIES
The aristotype cubic perovskite structure is usually
adopted at high temperature, while at low temperature
an assortment of lower symmetry phases (e.g. tetragonal,
orthorhombic, monoclinic, rhombohedral) are observed.
The lattice distortions associated with these transitions
can be divided into three categories: (i) polar displace-
ment of the A or B cations away from their high symme-
try positions; (ii) rigid tilting of the corner-sharing BX
6
octahedra; and (iii) collective Jahn-Teller distortion of
the BX
6
octahedra.
23–26
Cation displacements are usually responsible for so-
called proper ferroelectricity, which is associated with the
presence of a soft polar phonon mode at the Γ point
in the phonon Brillouin Zone. For example, BaTiO
3
and PbTiO
3
exhibit spontaneous polarization due to
the displacement of the Ti atom from the center of its
octahedron.
27
The displacive phonon mode is at the Bril-
louin Zone center which implies an in-phase periodic dis-
tortion across the crystal and thus a macroscopic polar-
ization.
On the other hand, octahedral tilting due to zone-
boundary phonon modes (e.g. at the M (
1
2
,
1
2
, 0) or
R (
1
2
,
1
2
,
1
2
) special points in the first Brillouin Zone of
the cubic structure) result in an anti-phase periodic dis-
tortion. The opposite polarization generated in neigh-
bouring unit cells cancels out and there is no resulting
macroscopic polarization. Due to this, these are often
referred to as antiferroelectric or antipolar distortions.
In-phase tilting corresponds to an M
+
3
mode denoted by
+ in Glazer’s notation a
0
a
0
c
+
, whereas out-of-phase tilt-
ing corresponds to an R
+
4
mode denoted by a − sign in
a
0
a
0
c
−
.
28
A linear combination of both modes can de-
fine all the possible rigid tilting modes in the perovskite
systems.
24
We previously explored the behaviour of the
M
+
3
mode in the caesium lead and tin halides.
29
A third type of distortion is due to the Jahn-Teller (JT)
effect. The octahedra distort by elongation or shorten-
ing of the B-X bond, and sometimes by off-centering of
the B cation. A first-order JT distortion is commonly
observed in transition metals with degenerate electronic
ground states such as Cu(II). Shortening/elongating the
nearest-neighbor bonds and generating a crystal field lifts
the degeneracy of the partially occupied d band (e.g. d
9
for Cu
2+
in KCuF
3
) and allows for a lowering of the en-
ergy. Cases such as Pb(II) 6s
2
where the structural dis-
tortion allows mixing with nominally empty 6s
0
orbitals
are usually referred to as second-order JT distortions.
30
These classes of distortion are not necessarily inde-
pendent, but can couple via (anharmonic) interactons
between phonon modes. Benedek et al. found that
this coupling acts to suppress ferroelectricity in many
oxide perovskites.
25
Moreover, there has been large in-
terest in designing new “hybrid improper” ferroelec-
tricity through coupling between non-polar modes in
Ruddlesden-Popper phase perovskites.
31
In particular,
tilting coupled with Jahn-Teller distortions (so-called
“pseudo” rotations) are considered one route for design-
ing multiferroic materials.
32–34
III. METHODS
Model structures for 24 inorganic halide compounds
ABX
3
(A = Cs, Rb; B = Ge, Sn, Pb; X = F, Cl,
Br, I) were taken from the Inorganic Crystal Struc-
ture Database where available, and the remaining gen-
erated by atomic substitution of similar structures. Den-
sity functional theory (DFT) as implemented in the
pseudopotential plane-wave code VASP
35,36
was used
to optimize the crystal structures of four commonly-
observed phases viz. cubic P m
¯
3m, tetragonal P4/mmm,
orthorhombic P nma and monoclinic P 2
1
/m. The struc-
tures were fully relaxed with the PBEsol exchange-
correlation functional.
37–39
Explicit convergence testing
identified a plane-wave kinetic-energy cutoff of 800 eV
and an electronic Brillouin zone sampling with an 8×8×8
mesh for the cubic and orthorhombic phases, a k-point
mesh of 6×6×8 mesh for the tetragonal phases, and
a 4×4×4 mesh monoclinic phases to produce accurate
phonon frequencies.
The formation enthalpy of each crystal can be calcu-
lated from
∆H
ABX
3
= E
ABX
3
− µ
A
− µ
B
− 3µ
X
(1)
where E
ABX
3
is the total energy obtained in DFT cal-
culation, µ are the chemical potentials of the constituent
elements. When comparing the differences in ∆H
ABX
3
between different phases with the same composition, the
µ terms cancel and ∆H is simply the difference in DFT
total energies for the different phases.
Harmonic phonon dispersion and density of states
(DoS) curves were obtained using the finite displacement
method implemented in the open-source Phonopy pack-
age, with VASP used as the force calculator. For each
phase, a series of symmetry-independent displacements
were generated in 2×2×2 supercell expansions, chosen
to be commensurate with the zone-boundary symmetry
points, to obtain the force–constant matrices, which were
then used to obtain phonon frequencies and eigenvectors
at arbitrary phonon wavevectors q.
To map out the anharmonic potential energy surfaces
associated with the phonon instabilities, we use the open-
3
FIG. 1. a) Schematic potential energy surface along harmonic
and anharmonic phonon modes as a function of the phonon
normal mode coordinate Q. b) Schematic illustration of the
three classes of distortion observed in cubic perovskites and
the associated irreducible representations from group theory.
The ferroelectric displacement is associated with a zone-centre
phonon mode that leads to macroscopic electric polarization.
The octahedral tilting is associated with a zone-boundary
phonon mode and results in an expansion of the unit cell.
Jahn-Teller distortions can be associated with either a zone-
centre or zone-boundary mode, but in the cubic perovskites
studied here manifest as zone-boundary M or X modes. These
distortions may coexist in real materials.
source ModeMap package.
40,41
A sequence of displaced
structures in a commensurate supercell expansion are
generated by displacing along the phonon eigenvectors
over a range of amplitudes of the normal-mode coordi-
nate Q, and the total energies of the “frozen phonon”
structures are evaluated from single-point DFT calcu-
lations. The E(Q) curves are then fitted to a polyno-
mial function, with the number of terms depending on
the form of the potential well. These anharmonic energy
surfaces then effectively include higher-order terms in the
potential energy as a function of the nuclear coordinates,
in the basis of the harmonic eigenvectors.
IV. PHASE DIVERSITY OF INORGANIC HALIDE
PEROVSKITES
A search of the literature shows that only a fraction
of the 24 chemical combinations studied here have been
reported experimentally, as summarised in Table I. The
majority of the inorganic halide perovskites are reported
to adopt the cubic structure (space group P m
¯
3m) at high
temperature, including CsGeCl
3
, CsPbCl
3
, CsPbBr
3
,
CsPbI
3
, CsSnI
3
, RbGeI
3
, RbPbF
3
,
42–48
while CsSnBr
3
has been reported to be cubic at room temperature.
2
The phase diversity can be qualitatively explained us-
ing the concept of the tolerance factor α introduced by
Goldschmidt
49
where:
α = (r
A
+ r
X
)/
√
2(r
B
+ r
X
) (2)
Values of α < 1 are usually associated with octahedral
tilting due to an A cation that is smaller than the ideal
value for the BX
3
octahedral framework. This is the
case for the majority of compounds considered here (see
Table I), which explains the most stable phases being
lower-symmetry space groups, typically Pnma, which is
the ground state structure of CsPbCl
3
, CsSnI
3
, RbPbF
3
,
RbPbI
3
.
43,45,46,48
However, many of the Ge compounds have α > 1,
which often leads to B-site displacements and hence to
polar structures due to loss of centrosymmetry. For in-
stance, CsGeCl
3
, CsGeBr
3
and CsGeI
3
are reported to
undergo an order-disorder phase transition from cubic to
rhombehedral (R3m) under ambient conditions.
4,50
We computed the formation enthalpy and phonon dis-
persion of four phases for each composition (see Support-
ing Information). Imaginary frequencies (soft modes)
present in the phonon dispersion indicate dynamic insta-
bilities at 0 K, whereas the absence of imaginary modes
indicates the structure is a local potential-energy mini-
mum. The stable phases of each composition are sum-
marised in Table I. None are found to be most sta-
ble in the cubic or tetragonal phases, with the corre-
sponding dispersion curves showing soft modes across the
whole Brillouin zone, indicating multiple dynamical in-
stabilities. Similarly, the tetragonal phases, derived from
the cubic structure by in-phase octahedral rotation with
Glazer notation a
0
a
0
c
+
, also show multiple instabilities.
This indicates that the tetragonal phases are intermedi-
ates formed on the symmetry lowering path from cubic
to the lowest-energy symmetry-broken ground state.
The lower-symmetry phases of all compositions have
more favourable formation enthalpies than the cubic
phase, and the orthorhombic P nma and monoclinic
P 2
1
/m phase also have lower formation energies than
tetragonal P 4/mmm structure, showing that additional
distortions lower the energy (Table I). This shows that
4
the tetragonal phases are both dynamically and energeti-
cally unstable at 0 K. However, the formation enthalpies
of the P nma and P 2
1
/m phases are similar, implying
that the energy gain for further lowering the symmetry
from orthorhombic to monoclinic is small. Consideration
of the formation enthalpy differences can help to explain
why the orthorhombic phase is the most commonly ob-
served for halide perovskites.
V. HARMONIC PHONON INSTABILITIES
In the cubic perovskite phase, all compounds were
found to exhibit imaginary modes at certain symme-
try points in the phonon Brillouin zone (Table II). All
compounds exhibit M-point instabilities. Apart from
CsSnBr
3
, all the compounds also exhibit Γ-point insta-
bilities. Perovskites other than CsPbF
3
, CsSnBr
3
and
CsSnCl
3
also show X-point instabilities. In addition, R-
point instabilities are found in all perovskites other than
CsGeX
3
(X = F, Cl, Br) and CsSnF
3
. It is worth noting
that all the Rb compounds feature instabilities at every
high-symmetry point in the cubic perovskite Brillouin
zone, in agreement with the fact that fewer Rb halide
perovskites have been identified experimentally.
To provide a better understanding of the underly-
ing chemical driving forces, we have categorized the in-
stabilities into zone-centre cation displacements, zone-
boundary octahedral tilting, and Jahn-Teller distortions
(Fig. 1). We will show in the following sections how
the chemistry of the compound determines the types of
instabilities that occur.
A. Octahedral tilting
Instabilities corresponding to the octahedral tilting can
be found at both the M and R points. Among the
24 cubic halide perovksites all show M-point instabili-
ties, allowing comparisons to be made between the sys-
tems. Although the imaginary modes occur at the same
reciprocal-space wavevector, they can correspond to dif-
ferent types of distortions in real space. The harmonic
eigenvectors associated with these modes were therefore
inspected to determine the distortion type. Some com-
pounds have more than one M-point instability, but for
simplicity we confine our discussion to the “most” imag-
inary phonon branch, i.e. the one lying lowest in the
phonon dispersion, which is indicative of the largest neg-
ative curvature of the potential-energy surface around Q
= 0.
CsSnCl
3
, CsSnBr
3
, CsSnI
3
, CsPbX
3
, RbGeBr
3
,
RbGeI
3
, RbSnCl
3
, RbSnBr
3
, RbSnI
3
and RbPbX
3
dis-
play M
+
3
instabilities corresponding to an in-phase oc-
tahedral tilting with Glazer notation a
0
a
0
c
+
(Fig. 3).
For CsGeX
3
, CsSnF
3
, RbGeF
3
, RbGeCl
3
, RbSnF
3
, the
lowest-lying phonon mode corresponds to the M
−
2
mode,
which corresponds to a Jahn-Teller distortion and will be
discussed in the following section.
Using the eigenvectors associated with the M
+
3
mode,
we mapped out the potential energy surface E as a func-
tion of the normal-mode coordinate Q. The E −Q curves
for M
+
3
in all compositions that adopt this instability are
shown in Fig. 2. The resulting E−Q surfaces correspond
to symmetrical double-well potentials, with the central
point Q = 0 being the reference cubic phase. The well
depth indicates the enthalpy gained by breaking the crys-
tal symmetry. Shallow well depths (∆E ≈ k
B
T ) suggest
that thermal energy can allow the system to explore parts
of the potential close to Q = 0. Such anharmonic double
wells are common in perovskite structures
59,60
and are
characteristic of ferroelectric transitions for cases where
Q represents a polar distortion.
61,62
The Rb compounds possess deeper minima than their
Cs counterparts (Fig. 2), indicating a stronger energetic
driving force for tilting. Within the Rb or Cs series,
there are two distinct trends: (i) the double well depth
increases monotonically when the B-site cation goes from
Ge → Sn → Pb; (ii) for compounds with the same B-site
cation, the well depth increases as the X site is substi-
tuted from F → Cl → Br → I.
Both phenomena can be interpreted by the geometry
of the cubic perovskite crystal. Firstly, the Rb cation
is too small for the octahedral cavity, which drives the
collapse of the network towards the A cation. Secondly,
the interplay between the size of the A-site cation and
the volume of the cage also determines the magnitude
of the tilting instability, i.e. the larger the ratio of the
size of the cavity to the A-site cation, the stronger the
tilting. This again follows the tolerance factor α. If α
is too small, the octahedral network will tend to tilt, as
observed in RbPbX
3
, which have the smallest α (0.82
– 0.86) and the largest well depths. On the other hand,
when α is close to or greater than one, such as in CsGeX
3
(0.98–1.11), RbGeF
3
(1.05) and RbGeCl
3
(0.98), the M
+
3
tilting instability is not seen due to the limited space in
the cuboctahedral cavity.
An almost-identical trend is observed for the out-of-
phase tilting R
+
4
mode, which suggests that the behavior
is not limited to in-phase tilting but is common to all
tilting modes.
B. Zone-boundary Jahn-Teller distortion
In addition to rigid octahedral tilting, some com-
pounds show M-point imaginary frequencies associated
with a different type of distortion. For most of the
Ge perovskites and some of the Sn perovskites, in-
cluding CsGeF
3
, CsGeCl
3
, CsGeBr
3
, CsGeI
3
, RbGeF
3
,
RbGeCl
3
, CsSnF
3
, and RbSnF
3
, the imaginary modes
with the steepest local curvature correspond to an M
−
2
instability (Fig. 4). Instead of rigid rotation of the oc-
tahedra with fixed bond lengths, the B–X bonds either
shorten or lengthen and the B-site cation is displaced
5
TABLE I. Difference in formation enthalpy (∆H) for tetragonal (P 4/mbm, a
0
a
0
c
+
), orthorhombic (P nma, a
+
b
−
b
−
) and
monoclinic (P 2
1
/m, a
+
b
−
c
−
) phases of the ABX
3
perovskites with respect to the cubic phase (P m
¯
3m, a
0
a
0
a
0
). Phases
identified as being phonon stable and experimentally-observed structures are also given. The tolerance factor is calculated
using the Shannon ionic radii where available
51
, while the radius for Sn(II) (1.15
˚
A) is taken from Ref. 52.
Composition ∆H (meV/formula) Phonon stable phase Experimentally observed Tolerance factor
P 4/mbm P nma P 2
1
/m
CsGeF
3
-0.9 -1 -0.7 1.11
CsGeCl
3
-0.3 -1 -0.3 P m
¯
3m at 449 K/ R3m at RT
53
1.02
CsGeBr
3
-2.6 -4.0 -1.1 R3m at RT
4
1.01
CsGeI
3
-2.6 -4.4 -0.2 R3m at RT
4
0.98
CsSnF
3
0.2 -1.8 -2.5 0.92
CsSnCl
3
-0.9 -27.7 -27.3 P m
¯
3m at RT
54
0.88
CsSnBr
3
-22.8 -35.9 -35 P nma and P 2
1
/m P m
¯
3m at RT, P nma at 100 K
2,55
0.87
CsSnI
3
-634.1 -692.9 -691.8 P 2
1
/m P m
¯
3m at 500 K; P nma at RT
46
0.86
CsPbF
3
-20.4 -32 -32 P nma and P 2
1
/m P m
¯
3m at 186 K; R3c at RT
56,57
0.90
CsPbCl
3
-63.8 -85.5 -85.3 P m
¯
3m at 320 K; P nma at 310 K
15
0.87
CsPbBr
3
-72.5 -100.5 -100.4 P 2
1
/m P m
¯
3m at 403 K ; P nma at 361 K
58
0.86
CsPbI
3
-94.4 -137.5 -137.3 P nma P m
¯
3m at 634 K
45
0.85
RbGeF
3
-0.7 -0.1 -0.7 1.05
RbGeCl
3
-8.2 -1.8 -12.3 0.98
RbGeBr
3
-17.8 -22.4 -26.5 0.97
RbGeI
3
-41.3 -47.4 -65.2 P m
¯
3m at 533 K
47
0.95
RbSnF
3
-73.2 -97.9 -97.9 0.87
RbSnCl
3
-105.4 -156 -155.8 0.84
RbSnBr
3
-103.7 -158.2 -157.7 P nma and P 2
1
/m 0.84
RbSnI
3
-117.5 -185.9 -185.7 P nma and P 2
1
/m 0.83
RbPbF
3
-134.9 -188.4 -188.3 P nma and P 2
1
/m P m
¯
3m at 515 K ; P nma at RT
48
0.86
RbPbCl
3
-170.4 -249.2 -196.7 P nma 0.83
RbPbBr
3
-171.3 -255.2 -197.1 P nma and P 2
1
/m 0.83
RbPbI
3
-185.9 -289.4 -289.4 P nma and P 2
1
/m P m
¯
3m at RT
45
0.82
FIG. 2. Double-well potential-energy surfaces for the phonon instabilities associated with M
+
3
tilting modes. The left-hand
plot shows data for Cs compounds (lines with blue hue), while the right-hand plot shows data for Rb perovskites (lines with
red hue). Halide anions are differentiated by the markers shown in the legends. The normal-mode coordinate Q has been
normalised so that for each composition the minima are located at Q = 1, and the energy differences are those calculated in a
2×2×2 supercell.
