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# Cation Dynamics in Hybrid Halide Perovskites

Eve M. Mozur  1 James R. Neilson  1 1Chemistry Department, Colorado State
University, Fort Collins CO, USA, 80523; email<EMAIL_ADDRESS>
###### Abstract

Hybrid halide perovskite semiconductors exhibit complex, dynamical disorder
while also harboring properties ideal for optoelectronic applications that
include photovoltaics. However, these materials are structurally and
compositionally distinct from “traditional” compound semiconductors composed
of tetrahedrally-coordinated elements with an average valence electron count
of silicon. As discussed here, the additional dynamic degrees of freedom of
hybrid halide perovskites underlie many of their potentially transformative
physical properties. Neutron scattering and spectroscopy studies of the atomic
dynamics of these materials have yielded significant insights to the
functional properties. Specifically, inelastic neutron scattering has been
used to elucidate the phonon band structure, and quasi-elastic neutron
scattering (QENS) has revealed the nature of the uncorrelated dynamics
pertaining to molecular reorientations. Understanding the dynamics of these
complex semiconductors has elucidated the temperature-dependent phase
stability and origins of the defect-tolerant electronic transport from the
highly polarizable dielectric response. Furthermore, the dynamic degrees of
freedom of the hybrid perovskites provides additional opportunities for
application engineering and innovation.

###### doi:

10.1146/((please add article doi))

###### keywords:

hybrid perovskite, neutron scattering, semiconductor, spectroscopy

††journal: Xxxx. Xxx. Xxx. Xxx.

###### Contents

  1. 1 INTRODUCTION: HYBRID HALIDE PEROVSKITE SEMICONDUCTORS
  2. 2 NEUTRON SCATTERING, A BRIEF PRIMER
  3. 3 QUASI-ELASTIC NEUTRON SCATTERING (QENS), A CLOSER LOOK
  4. 4 BACKGROUND: DYNAMICS AND ELECTRONIC PROPERTIES
    1. 4.1 Collective, Periodic Dynamics: Phonons
    2. 4.2 Independent, Aperiodic Dynamics
      1. 4.2.1 Inorganic dipolar dynamics
      2. 4.2.2 Organic cation dynamics
  5. 5 CASE STUDIES IN HYBRID PEROVSKITE SEMICONDUCTORS
    1. 5.1 Unsubstituted Hybrid Halide Perovskite
      1. 5.1.1 Methylammonium Lead Halide Perovskites
      2. 5.1.2 Formamidinium Lead Perovskite Halides
      3. 5.1.3 Lead-free Hybrid Perovskite Halides
    2. 5.2 Effects of Chemical Substitution
      1. 5.2.1 Cesium Substitution in Methylammonium Lead Bromide
      2. 5.2.2 Cesium Substitution in Formamidinium Lead Bromide
      3. 5.2.3 Complex, Multiple Substitutions in Hybrid Perovskite Halides
  6. 6 CONCLUSIONS AND OUTLOOK

##  1 INTRODUCTION: HYBRID HALIDE PEROVSKITE SEMICONDUCTORS

Hybrid halide perovskites and their structural derivatives have advantageous
optoelectronic properties for a relatively wide range of applications. The
tunable band gap, defect tolerance and long electronic excited state lifetimes
of these perovskites make them ideal semiconductors for photovoltaics [1, 2].
The heavy elements in these materials also allows their use in X-ray and
$\gamma$-ray detection [3, 4]. Changing the composition or reducing the
effective dimensionality changes the electronic structure and exciton binding
energy to increase the efficiency of light emission; several materials even
reemit white light intrinsically [5]. This family of materials provides new
opportunities for building the structure-property relationships of next-
generation semiconductors.

Figure 1: (a) Schematic illustration of the plastic crystalline nature of
hybrid perovskites of the formula, $ABX_{3}$, as well as the principal
rotations of methylammonium, CH3NH_3^+, and formamidinium, CH(NH2)_2^+. (b)
Cartoon illustrating the scale of the various ions found in hybrid halide
perovskites based upon their Shannon-Prewitt radii [6] or estimated molecular
size [7]. (c) Commonly proposed electron-phonon interactions in hybrid halide
perovskites involving organic cation dynamics, as described in the text.

Hybrid halide perovskites exhibit additional dynamic degrees of freedom when
compared to traditional compound semiconductors (e.g., CdTe, GaAs, TlBr).
Materials such as (CH3NH3)PbI3 and (CH(NH2)2)PbI3 are plastic crystals, as
schematically depicted in  Figure 1(a): the organic cations exhibit
translational symmetry with respect to their center of mass but without
orientational order[8]. The molecular ‘sphere’ approximated by rotating the
molecules can be used to compute a revised Goldschmidt tolerance factor based
on the ionic radii of the constituent ions (Figure 1(b)) [7], as often used to
predict the formation of a crystal structure analogous to CaTiO3, the mineral
Perovskite [9]. Similar to oxide perovskites, the halide perovskites typically
undergo phase transitions on cooling, as characterized by a reduction in
symmetry from cubic to tetragonal to orthorhombic related to cooperative
tilting of the octahedral framework [10, 11]. However, the rich and subtle
details of these transitions and the resulting properties of each phase cannot
be fully understood without considering the plastic crystalline nature of
these materials [12].

Various advantageous electronic properties in hybrid perovskites have been
attributed to the dynamic, plastic crystalline behavior. While long electronic
excited state lifetimes have been linked to collective dynamics of the
extended octahedral framework (i.e., phonons) [13, 14, 15, 16, 17], such long-
lived excited states are also linked to organic cation reorientational
dynamics by way of giant polaron [18], phonon bottleneck [19], or Rashba
splitting [20] based mechanisms (Figure 1(c); described in more detail in
Section 4.2.2). Furthermore, the presence of an organic cation has been
associated with the “self healing” of photoinduced defects [21] as well as in
aiding in photoinduced ionic conductivity [22, 23, 24]. The ionic mobility of
hybrid halide perovskites is often linked to decomposition [25, 26, 27, 28]
and is hypothesized to relate to stationary atomic fluctuations [29], akin to
the “paddlewheel” mechanism in materials studied for alkali ion conduction
[30, 31]. Similar dynamics are thought to influence relaxation pathways in
layered perovskite derivatives of interest as phosphors [32, 33]; the
microscopic details remain an active area of research. Regardless of the
proposed mechanism, the impressive “defect tolerant” properties of hybrid
halide perovskites advance high-efficiency photovoltaics [34, 35, 1]. This
review focuses on how neutron scattering, particularly quasi-elastic neutron
scattering, has addressed questions related to these properties by
characterizing the dynamic nature of organic cations in hybrid perovskite
semiconductors.

Phonon Bottleneck A bottleneck of heat-transporting phonons prevents
electronic relaxation, such as the relaxation of “hot” carriers to the band
edges [36]. Rashba splitting Broken inversion symmetry in the presence of
strong spin-orbit coupling splits doubly degenerate bands into spin-up vs
spin-down bands in momentum space to yield a material with close direct and
indirect transitions, enabling strong absorption but long-lived carriers.

[ht]

##  2 NEUTRON SCATTERING, A BRIEF PRIMER

Neutron scattering is a powerful tool to investigate the structure and
dynamics of materials through analysis of the dynamic structure factor,
$S(\bm{Q},\omega)$ (Figure 2(a)). The large mass of neutrons, compared to
photons or electrons, means that neutrons have a relatively large momentum
that allows them to couple strongly to atomic and molecular dynamics. The
scattering cross section depends on interactions of the neutron with the
nucleii within each atom and is therefore isotope dependent (Figure
2(b),[37]). The lack of monotonic trends in cross section with atomic number
also provides complementary contrast to X-ray scattering techniques, and the
large (incoherent) cross section of ^1H makes neutron spectroscopy ideal for
studying hybrid perovskites. The modulation of a neutron’s speed through a
moderator once released from either a reactor or spallation source allows for
tunability of incident energy. Therefore, low wavelengths (0.5 to 4 Å) are
suited for atomic-resolution diffraction, while their energies ($\mu$eV-eV)
are ideal for spectroscopy of atomic and molecular dynamics. The neutron
scattering techniques most relevant for investigating the structure and
dynamics of solid state materials include neutron diffraction, quasi-elastic
neutron scattering (QENS), and inelastic neutron scattering (INS), as
summarized in Figure 2. Diffraction experiments are not typically energy
resolved; therefore, they characterize predominantly elastic interactions that
inform the average structure (intensity of elastic to inelastic scattering
events $\sim$100:1). INS probes the phonon and molecular vibrational density
of states between $\sim$5 meV to 1000 meV. QENS probes dynamics on the $\mu$eV
energy scale, encompassing molecular reorientations, translations, and
diffusion, as described in the next breakout panel. These dynamics map onto
the dielectric response function illustrated in Figure 2(e,f).

Currently, these experiments can only be conducted at large, shared
spallation- or reactor-based neutron sources. As such, INS is complementary to
infrared and Raman spectroscopy; QENS is complementary to other techniques
sensitive to motions on $10^{-9}$ s to $10^{-12}$ s time scales, such as
nuclear magnetic spectroscopy (also nucleus specific) and optical
spectroscopies (e.g., Brillouin light scattering [38] or ultrafast vibrational
spectroscopy[39]). However, the large neutron scattering cross section of
hydrogen makes QENS ideal for studying the molecular dynamics in hybrid halide
perovskites.

[ht]

##  3 QUASI-ELASTIC NEUTRON SCATTERING (QENS), A CLOSER LOOK

Quasi-elastic neutron scattering (QENS) can be performed on neutron
spectrometers with excellent energy resolution ($\Delta E\sim$ 1 to 3
$\mu$eV). The quasi-elastic scattering (e.g., energy transfer $\neq 0$ within
the instrumental resolution) yields information about slow motions
corresponding to relaxation of nearly stationary states, such as diffusion or
molecular relaxation via translations or rotations. These relaxations provide
a Doppler broadening of the elastic scattering due to the small energy
transfer between neutrons and the moving nuclei in the sample. This is often
differentiated from inelastic scattering, where the neutrons excite or are
excited by discrete motions with finite energies (e.g., internal molecular
vibrations).

To analyze the molecular motions, one can use a “jump model,” as molecules are
assumed to reside in preferred orientations with frequent (e.g., GHz) jumps
between symmetry-related (site and molecular) orientations [40]. The quasi-
elastic linewidth relates to the timescale of this residence time between
jumps (Figure 2(a), Lorentzian half width at half maximum = $\hbar/\tau$).
However, motions with a frequency faster than the maximum energy range of the
spectrometer (anywhere from $\hbar\omega_{max}$ = $\pm$30 $\mu$eV to $\pm$1000
$\mu$eV, depending on instrument configuration) appear as a background and are
not detected. Since one typically measures the incoherent structure factor,
$S_{\text{inc}}(Q,\omega)$, the $Q$-dependence of the elastic scattering
relative to the sum of elastic and quasi-elastic scattering (i.e., the elastic
incoherent structure factor, EISF
$=I_{\text{elastic}}/(I_{\text{elastic}}+I_{\text{quasielastic}})$ ) can be
fit with a jump model to extract the type of motion, including: the local
symmetry of motion, distance of motion, and geometry of object moving, as
illustrated in Figure 2(c).

With such excellent energy resolution on a neutron spectrometer, one can
accurately measure the fraction of elastic scattering. The incoherent elastic
scattering yields information about thermal motion, which follows a
$Q$-dependent attenuation depending on the mean squared displacement (MSD) of
atomic motion via the the Debye-Waller factor = $\exp(-q^{2}\langle
u^{2}\rangle/3)$, for isotropic, harmonic motion, as shown in Figure 2(d).
These experiments can be accomplished quickly on some instruments, thus
permitting parametric studies, as to follow the temperature dependence of
thermal motion through phase transitions.

Figure 2: (a) Schematic representation of features observable in neutron
spectroscopy, including elastic scattering from static atoms, quasi-elastic
neutron scattering (QENS) from molecular reorientational dynamics, and
inelastic neutron scattering from quantized lattice vibrations (acoustic and
optical phonons) and molecular vibrations. (b) Neutron scattering cross
section of different natural-abundance averaged elements [37], highlighting
the large cross section of ^1H relative to the other elements germane to
hybrid perovskites. (c) The elastic incoherent structure factor (EISF) used to
extract details of molecular motion from momentum transfer dependence of the
QENS, illustrated by three different jump models corresponding to $C_{2}$,
$C_{3}$, $C_{4}$ rotations of the same hypothetical molecule. Grey squares
illustrate hypothetical data at discrete values of $Q$ from one sample at
fixed temperature. Different jump models can be challenging to discriminate
without a sufficient $Q$-range. (d) Schematic illustration of the incoherent
elastic scattering as a function of momentum transfer, $S_{inc}(Q,\omega=0)$,
used to extract the neutron scattering cross-section weighted atomic mean-
squared displacement, $\langle u^{2}\rangle$ = 0.5 Å2. Grey squares illustrate
hypothetical data at discrete values of $Q$ from one sample at fixed
temperature. (e,f) Dielectric function, $\epsilon$, including the real (solid
line), imaginary (dashed), and $\Im(-1/\epsilon)$ (dotted) components of (e)
CH3NH3PbI3 [41, 42] at 298 K as compared to (f) molecular, liquid H2O at 298 K
[43] and paraelectric ceramic BaTiO3 at 1000 K [44]. The frequency-dependent
dielectric function highlights contributions from either molecules (beige
shading) or phonons (lavender shading), as detectable by (a) neutron
scattering.

##  4 BACKGROUND: DYNAMICS AND ELECTRONIC PROPERTIES

The frontier electronic states of halide perovskites couple primarily to the
low energy optical phonons. The influence of lattice dynamics on
semiconductors typically revolves around a discussion of the scattering of
charge carriers from electron-phonon coupling. In compound semiconductors with
the diamond-type structure such as GaAs, optical phonons reside near $\sim$40
meV; thus they are not appreciably populated at room temperature and play a
relatively small role in electron-phonon scattering processes [45]. In
contrast, halide perovskite optical phonons reside from $\sim$5 meV to
$\sim$12 meV (Figure 2). Therefore, a complete understanding of the charge
transport necessitates characterization of the electron-phonon coupling. In
the following section, we provide a brief overview of the importance of these
collective dynamics in the context of light-absorbing hybrid semiconductors
for photovoltaics, and then we discuss the presence and role of localized
(i.e., aperiodic) dynamics and their proposed roles in influencing the
optoelectronic performance of hybrid perovskites and related semiconductors.

ExcitonUsed here as the Coulombically-attracted bound state of negative and
positive charge carriers in a crystal. Fröhlich interactionCoupling of a
charge to the presence of the electric field generated from a longitudinal
optical phonon [46]. PolaronThe distortion of an ionic lattice in response to
extra charges, akin to the Fröhlich mechanism.

###  4.1 Collective, Periodic Dynamics: Phonons

Optical phonons, primarily derived from the cooperative dynamics within the
octahedral framework, drive the phase behavior of halide perovskites. Akin to
the ferroelectric perovskite oxides with symmetry-lowering phase transitions
on cooling from a paraelectric phase [47], the phonons related to these phase
transitions contribute a significant dielectric response, as in the case of
BaTiO3 (Figure 2(f)). Perovskite halides undergo similar symmetry-lowering
phase transitions involving changes to the octahedral tilting upon cooling,
which relate to soft phonon modes [48, 49, 50]. Significant entropy releases
accompany these canonical phase transitions [51], as related to the loss of
orientational degrees of freedom of the organic molecule within the changing
shape of the nominally cuboctahedral $A$-site void [52, 53, 54, 55].

Coherent scatteringYields structural and dynamical structural information.
Incoherent scatteringYields information independent of positional
correlations.

These same collective dynamics influence charge transport primarily through
electron-phonon coupling. The optical phonons generate just over half of the
polarizability in the dielectric response of CH3NH3PbI3, as portrayed in
Figure 2(e). In halide perovskites, electron-phonon scattering via the
Fröhlich interaction limits the mobility of both free carriers and excitons;
therefore, the Fröhlich interaction is the most important form of electron-
phonon coupling [15, 16, 14]. The magnitude of the Fröhlich interaction in
CH3NH3PbI3 ($\alpha=$ 1.7-2.4 [18]) derives predominantly from a $\sim$11 meV
longitudinal optic (LO) phonon (see Figure 2(e)) and compares to that of other
halide semiconductors, such as TlBr ($\alpha=$2.05); this contrasts with the
smaller coupling constants of diamond-lattice, tetrahedral semiconductors GaAs
($\alpha=$0.068) and CdTe ($\alpha=$0.39) [42, 17].

###  4.2 Independent, Aperiodic Dynamics

####  4.2.1 Inorganic dipolar dynamics

The fluctuations of lone pair electrons modify the structure and properties of
hybrid perovskites. They have been linked to many exciting aspects of hybrid
perovskites, including the large ionic dielectric responses, light holes, and
unconventional band gap: all contribute to the favorable electronic transport
and “defect tolerance” [56]. These “inorganic dipoles” [57] resemble the
behavior of plastic crystals with permanent, molecular dipoles (e.g.,
potassium cyanide [58]), but with dynamics on a fast $10^{14}$ Hz time scale
rather than the slower $\sim 10^{12}$ Hz time scale that is associated with
the ionic dynamics portrayed in Figure 2(e,f).

Lone Pair ElectronsGe(II), Sn(II), Pb(II), Sb(III), and Bi(III), have the
valence electron configuration, “$ns^{2}$,” that can become stereochemically
active to create space-filling lone pair electrons [59], as employed to
describe the nearly tetrahedral molecular geometry of NH3.

All-inorganic perovskites exhibit significant polar dynamics from anharmonic
ionic fluctuations, attributed to a “head-to-head” motion of cesium ions [60].
Such polar fluctuations and deformations of the $[BX_{6}]$ octahedra have been
implicated in the formation of polarons, which screen excited state charge
carriers that increase the excited state lifetimes [61, 62]. This explains how
all inorganic halide perovskites yield comparable transient electronic
responses and device performance to their “hybrid” relatives [63].

####  4.2.2 Organic cation dynamics

Hybrid halide perovskite are plastic crystals, and the dynamically disordered
organic cations impart new degrees of freedom relative to conventional
compound semiconductors. These dynamics play a role in various functional
properties, deriving primarily from their polarizability [64]. Nearly half of
the static dielectric polarizability of CH3NH3PbI3 derives from the dynamic
molecules, akin to that of a polar solvent such as water (Figure 2(e,f)). This
polarizability has several implications for charge transport, summarized in
Figure 1(c) and described below.

Although the exact mechanism behind the long excited state lifetimes for
thermalized and hot carriers remains debated, the orientational dynamics of
the organic molecules play an important role in many (and not mutually
exclusive) hypotheses. Organic cation dynamics have been implicated in the
formation of large polarons that interact with both exciton [65] and free
charge carriers [66, 67]. The presence of dynamic organic cations accelerates
the polaron formation time (0.3 ps in CH3NH3PbBr3 vs 0.7 ps CsPbBr3 [68]),
which extends the lifetime of hot carriers [69]. The coupling of inorganic
lattice distortions and organic cation reorientations has been shown to screen
charge carriers and excitons [13, 14]. For example, the lack of interactions
between independently reorienting molecules yields a low vibrational group
velocity, and thus poor thermal conductivity with a high heat capacity – an
ideal recipe for a “phonon bottleneck,” in which energetic carriers cannot
thermalize efficiently with the lattice due to poor heat removal, as detected
from pump-probe spectroscopy experiments [19, 70, 71] and supported by a
polaron model backed by density functional theory calculations [67].

Plastic CrystalsCharacterized by a lack of orientational order with a
preservation of translational order. Potassium cyanide is a canonical example,
where cyanide ions are orientationally disorded, but their center of rotation
has a well defined position. The dynamic disorder and the subsequent
temperature-dependent orientational ordering transitions of plastic crystals
give rise to numerous entropic, dielectric, and (ferro)elastic anomalies [8,
58, 72].

Furthermore, the presence of polar fluctuations has the potential to locally
and transiently break inversion symmetry of the bulk crystal. Given the slow
orientational time scale of the organic dynamics ($\sim 5-15$ ps, Figure 3)
relative to electronic dynamics, it has been proposed that Rashba spin-split
valence bands can give rise to a direct band gap-like efficient optical
absorption transitions[20, 73, 74, 75]; carrier thermalization leads to a
shifted momenta, thus resulting in an indirect-like recombination and longer
lifetime (Figure 1(c)). Experiments have concluded that the halide perovskites
can support static [76, 77] and dynamic [78, 79, 80, 81] Rashba spin
splitting, with recent evidence that the static effect only persists at low
temperatures in hybrids, and that both all-inorganic and hybrid materials host
a dynamic Rashba effect at room temperature [82]. The dynamic Rashba effect
appears to originate from anharmonic, polar thermal fluctuations of the
inorganic octahedra that may arise from lone pair electron activity and their
interactions with organic cation reorientations or cesium motions [82, 60, 61,
62]. Therefore, the interplay between organic- and inorganic-centered dynamics
has the potential to influence the electronic excited state dynamics.

The dynamic organic molecules also impact the ionic transport and diffusion.
Hybrid halide perovskites have a significant ionic contribution to the
conductivity under various circumstances [22, 83, 24, 84], including under
light illumination [23]. While ion motion leads to device hysteresis [22],
phase separation [85], and decomposition [23], motion of the halogens can lead
to the creation of short-lived electronic defects. Such rapid defect creation
can lead to defect densities above the thermodynamic limit with sufficiently
short lifetimes as to be benign to carrier scattering [86], further supported
by the self-elimination of intrinsic defects on cooling [87]. Furthermore, the
timescale of defect motion is faster in hybrid perovskites, as illustrated by
the relatively fast self-healing of CH3NH3PbBr3 and CH(NH2)2PbBr3 from laser-
induced defects relative to CsPbBr3 [21]. While this leads to a high tolerance
to unintentional defects, it makes controlled doping very challenging.

Given the importance of reorientational dynamics in hybrid perovskites for
their functional performance, many studies have sought to understand the
nature of these dynamics, both of simple hybrid halide perovskites and of
technologically-relevant complex alloys [88]. The remainder of this
perspective details several case studies of organic cation dynamics through
neutron scattering and complementary techniques.

##  5 CASE STUDIES IN HYBRID PEROVSKITE SEMICONDUCTORS

Figure 3:  Distribution of activation energies and residence times for organic
cation dynamics $A$Pb$X_{3}$ phases, as differentiated by halogen ($A$ =
methylammonium, formamidinium; $X$ = Cl, Br, I). (a) Activation energies for
molecular reorientational motion further differentiated by organic cation,
illustrated by box plots and swarms of individual data points. (b) Anisotropic
displacement parameters (ADPs) of halogen positions in cubic unit cells
($Pm\bar{3}m$) further differentiated by organic cation, illustrated by box
plots and swarms of individual data points. The ADPs are separated into those
perpendicular to ($U_{\perp}$) and parallel to ($U_{\parallel}$) the Pb-$X$
bonds. (c) Residence times at $T=300\pm 10$ K for either reorientational
(e.g., whole molecule tumble) or wobble (e.g., $C_{3}$ rotation about a C-N
axis in methylammonium) motions. This includes methylammonium and
formamidinium-based phases in their tetragonal and cubic phases. Box plots
illustrate the median (horizontal lines) and mean (box center), while the
markers illustrate data points. Data compiled from multiple references are
tabulated in the Supporting Information [54, 89, 90, 39, 91, 92, 93, 94, 95,
48, 52, 96, 97, 98, 99, 100, 101].

Myriad competing interactions govern the complex dynamics in hybrid
perovskites. The activation energies and residence times of the reorientations
of organic cations for several $A$Pb$X_{3}$ perovskites shown in Figure 3
reflect several trends in ionic size and hydrogen bonding strength. In phases
where reorientational dynamics are present (e.g., tetragonal and cubic; Figure
3(a)), the activation energy of that motion shows distinct trends for each
organic cation. In methylammonium perovskites, the activation energy stays
relatively constant as a function of halogen, which correlates more strongly
with the constant energy of hydrogen bonding across the halogen series
(0.26-0.27 eV/cell) [102] than with the changing size of the halogen (Figure
1). However, the nature of the halogen does influence the reorientational
barrier in formamidinium perovskites, where the increasing size of the halogen
leads to an increased cuboctahedral void size to allow for less inhibited
molecular reorientations. This also follows the expected trend in hydrogen
bonding strength: 0.16 eV/cell for Cl, 0.1 eV/cell for Br, and 0.09 eV/cell
for I [102]. The smaller size of methylammonium decreases the importance of
the cuboctahedral void size; the stronger dipole moment of methylammonium
yields a higher contribution of hydrogen bonding to the total energy [102].
However, the activation energy for methylammonium reorientations increases for
iodine, which correlates with an increase in the crystallographic atomic
displacement parameter of iodide perpendicular to the Pb-$X$ bond that
dynamically decreases the cuboctahedral void size (Figure 3(b)).

Additional complexity is apparent upon examination of the residence times of
organic molecules. For both methylammonium and formamidinium perovskites,
studies reveal two characteristic residence times for two distinct motions
identified as a long time-scale reorientation and a short time-scale wobble
(Figure 3(c), $\sim 17$ ps vs $\sim 0.33$ ps, respectively). These data
demonstrate the composition dependent interplay of hydrogen bonding and steric
interactions on each reorientational mode.

In this section, we present case studies of organic cation dynamics based on
neutron spectroscopy and complementary techniques (e.g., NMR, dielectric
spectroscopy, time-resolved vibrational spectroscopy), as well as their
dependence on chemical substitution, and their relationships with
optoelectronic properties. These case studies demonstrate the insight that
neutron scattering has to offer for hybrid halide perovskites.

###  5.1 Unsubstituted Hybrid Halide Perovskite

####  5.1.1 Methylammonium Lead Halide Perovskites

Figure 4:  (a) Inelastic neutron scattering ($Q$ integrated and powder
averaged) collected on the IN4 spectrometer from CH3NH3PbBr3 showing softening
of a $\sim$5 meV phonon as well as the replacement of a $\sim$11 meV
vibrational mode with a broad, quasi-elastic relaxation above the orthorhombic
to tetragonal phase transition. (b) Inelastic neutron scattering as
$S(Q,\omega)$ from powder CH3NH3PbBr3 showing the significant overdampening of
modes at $T=$ 200 K. The sharp, harmonic modes are replaced by quasi-elastic
fluctuations extending up to high energies. (c) MSD of CH3NH3PbBr3, $\langle
u^{2}\rangle$, extracted from an elastic window scan as a function of
temperature showing significant changes in hydrogen dynamics at the tetragonal
to orthorhombic phase transition, as well as below $T\approx$ 50 K. Reprinted
figure with permission from Swainson, I. P. Stock, C. Parker, S. F. Van Eijck,
L. Russina, M. Taylor, J. W. 2015. From Soft Harmonic Phonons to Fast
Relaxational Dynamics in CH3NH3PbBr3. Phys. Rev. B, 92 (10), 100303. Copyright
2015 by the American Physical Society.

Methylammonium lead halide perovskites have been the archetypal hybrid
perovskite for solar applications, due to the advantageous band gap of
methylammonium lead iodide and the ease of preparation of methylammonium lead
iodide compared to the cesium and formamidinium analogues [2, 103, 104]. Early
nuclear magnetic resonance (NMR) and calorimetric studies demonstrated that
methylammonium is dynamically disordered in the high temperature cubic phase,
with increased orientational ordering on cooling that relates to the two known
phase transitions [105]. Between 400 K and 100 K, methylammonium lead iodide
transitions from the high temperature cubic phase (space group: $Pm\bar{3}m$)
to a tetragonal phase (space group: $I4/mcm$) characterized by out-of-phase
octahedral rotations when looking perpendicular to the $c$ axis (Glazer
notation: $a^{0}a^{0}c^{-}$), and then to an orthorhombic phase (space group
$Pnma$) characterized by additional octahedral tilt axes (Glazer notation:
$a^{-}b^{+}a^{-}$) [105]. These transitions change the shape and size of the
cuboctahedral void where methylammonium sits and thus its dynamical behavior.

The phase behavior of CH3NH3Pb$X$3 perovskites reflects the presence of
organic-inorganic coupling. The phonon dispersion measured in the cubic phase
of methylammonium lead iodide with high energy resolution inelastic X-ray
(HERIX) scattering revealed large amplitude and anharmonic zone-edge
rotational instabilities of the octahedra and short-range disorder of the
octahedra [49]. This disorder is largest for the hybrids, as opposed to all-
inorganic materials [106]. Subsequent neutron scattering revealed that these
instabilities are resolution limited in energy, and thus correspond to static
(i.e., with lifetime of $\tau>36$ ps) tetragonal domains similar to the
critical scattering seen from SrTiO3 [50]. First principles calculations
reveal that these instabilities are strongly coupled to the librations and
reorientations of methylammonium [49].

In CH3NH3PbBr3, there is a clear correlation between the phonons of the Pb-Br
framework and methylammonium dynamics. At the orthorhombic to tetragonal
order-disorder transition, sharp modes attributed to methylammonium vibrations
decrease in scattered amplitude while heating; at the transition these modes
are replaced by broad QENS associated with methylammonium reorientation
(Figure 4(a)). The decrease in vibrational mode scattering amplitude
accompanies the softening of a $\sim$5 meV optical phonon from the Pb-Br
framework. Furthermore, the tetragonal phase phonons become strongly
overdamped when the organic cations become significantly dynamic (Figure
4(b)). In CH3NH3PbCl3 one observes overdampening of the framework phonon modes
across the same temperature scale as observed with bromine, but without the
order-disorder phase transition [107].

Given the significance of hydrogen bonding in CH3NH3Pb$X$_3 [102], the
organic-inorganic interactions are a likely source of anharmonic behavior, as
supported by density functional theory calculations of the rotational energy
barriers [97]. Organic-inorganic coupling in the form of so called “nodding
donkey” modes are observed in Raman and infrared spectroscopy experiments and
density functional theory-based calculations thereof [108, 109]. The vacancy
ordered double perovskites that lack covalent connectivity between octahedra
(e.g., (CH3NH3)2SnI6) also show anharmonic tilting behavior that is correlated
with the presence of organic cations [110]. Understanding this anharmonicity,
and its origins in the organic cation dynamics, is important for describing
locally-broken symmetries and describing the nature of electron-phonon
coupling beyond the linear response [111].

Figure 5: (a) QENS from CH3NH3PbI3. Neutron scattering intensity is shown as a
function of momentum ($Q$) and energy ($\hbar\omega$) transfers, measured at
370 K, 260 K, 180 K, and 130 K. Open white circles are the data, and the color
coded surface is the surface image of the calculated jump model. (b)
Relaxation time, $\tau$, of rotational modes of CH3NH3PbI3 obtained from
fitting QENS data at various temperatures and plotted against inverse
temperature ($1/T$). The colored solid lines are the fits to
$\ln(\tau)=E_{a}/(k_{b}T)-\ln(A)$, where $E_{a}$ is an activation energy in
units of meV per one molecule, $k_{B}$ the Boltzmann constant, and $A$ a pre-
exponential factor. The colored dotted lines are guides to the eye. The black
solid and dotted lines are temperature dependent dielectric permittivity
measured at 1 kHz by Onoda-Yamamuro, et al. [54], which show a sharp increase
at 160 K. (c) $C_{3}\otimes C_{4}$: illustration of 12 crystallographically-
equivalent sites for H atoms of CH3NH${}_{3}^{+}$ in the tetragonal
environment. $C_{3}$: 3 equivalent H atomic sites in the orthorhombic
environment. Reprinted figure with permission from T. Chen, B. Foley, B. Ipek,
M. Tyagi, J. R. D. Copely, C. M. Brown, J. J. Choi, S. H. Lee. 2015. Physical
Chemistry Chemical Physics 17 (46) 31278-31286. Copyright Royal Society of
Chemistry 2015.

Methylammonium rapidly reorients in the high-temperature cubic phase and
exhibits a decrease in amplitude, timescale, and dynamic degrees of freedom on
cooling as the cuboctahedral void constricts, as learned from extensive
neutron scattering and complementary NMR spectroscopy studies. In
methylammonium lead bromide, the temperature-dependent mean squared
displacements (MSDs) calculated from incoherent elastic neutron scattering
(Figure 4(c), [48]) capture the overall decrease in motion. The lack of
anomalies in $\langle u^{2}\rangle$ at the cubic to tetragonal phase
transition suggest that the entropy loss associated with this phase transition
(8.2 J mol-1 K-1, $\approx R\ln(2.7)$ [52]) correlates to a change in the type
of motion rather than a reduction in the spatial extent of methylammonium
dynamics, $\langle u^{2}\rangle$. In contrast, the clear feature in the MSD at
the tetragonal to orthorhombic phase transition suggests a discontinuous
change in how methylammonium reorients in the cuboctahedral void. These
observations are also consistent with the conservation of ^1H and ^14N NMR
lineshapes across the cubic to tetragonal phase transition and appreciable
narrowing after the tetragonal to orthorhombic phase transition [98, 90].

The reorientational dynamics of methylammonium are well described by a jump
model of transitions between idealized orientations, as originally proposed
from changes in degrees of freedom at various phase transitions determined
from calorimetry [52]. Rigorous group theoretical analysis by Chen, et al.
with a jump model provided a complete description of the QENS data (Figure 5,
[89]). This jump model involves the direct product of a $C_{3}$ rotation
(120∘) derived from the internal symmetry axis of methylammonium and a $C_{4}$
rotation (90∘) perpendicular to the $C_{3}$ rotation as imposed by the
symmetry of the lattice site point group symmetry. Both the $C_{3}$ and
$C_{4}$ modes are active in the cubic and tetragonal phases, with relaxation
times of $\sim 1$ ps and $\sim 5$ ps, respectively, as determined by two
different studies [89, 96]. This analysis of the QENS is consistent with
direct measurements of the molecular dynamics; two-dimensional transient IR
spectroscopy also reveals the presence of a fast (300 fs) wobbling-in-a-cone
motion similar to the $C_{3}$ rotation and a slow (3 ps) molecular
reorientation similar to the $C_{4}$ rotation [39].

The molecular reorientations correlate most strongly to the dielectric
response. As shown in Figure 5(b), the $C_{4}$ motion increases significantly
in relaxation time on cooling within the tetragonal phase, but this
reorientation is frozen out upon cooling into the orthorhombic phase.
Therefore, loss of the $C_{4}$ reorientation leads to a significant decrease
in the dielectric function [54, 112, 113, 41, 105], schematically depicted in
Figure 2.

Through this organic-inorganic coupling, the dynamic degrees of freedom of
methylammonium correlate to changes in the optoelectronic properties. For
example, the freezing out of the $C_{4}$ whole molecular reorientation at the
tetragonal-orthorhombic phase transition correlates to faster recombination of
excited state charge carriers as well as an increase in exciton binding
energies [114, 115]. These correlations suggest that the methylammonium
$C_{4}$ reorientations participate in the electron-phonon models described in
Figure 1.

####  5.1.2 Formamidinium Lead Perovskite Halides

Figure 6:  (a) Parameters from modelling the temperature-dependent
photoconductivity of CH(NH2)2PbBr3. Excitonic peak center (left axis, gray
circles), excitonic peak intensity (right axis, green squares), and interband
intensity (right axis, pink triangles). (b) Heat capacity data for
CH(NH2)2PbBr3. The crystallographically-observed phase transitions occur at
266 K and 153 K. (c) MSDs determined from fixed window elastic neutron
scattering of CH(NH2)2PbBr3 measured on heating (orange circles) and cooling
(blue squares). The colored, rectangular overlays illustrate the
cyrstallographic symmetry at each temperature, with dotted gray lines
indicating the crystallographic phase transition temperatures and dashed gray
lines showing other phase transition temperatures determined from heat
capacity and differential scanning calorimetry data. (d) EISFs extracted from
QENS spectra at T = 160 K, 155 K, and 130 K. For comparison, models of
$C_{2}$, $C_{3}$, and $C_{4}$ molecular rotations are shown in (e). Reprinted
with permission from Mozur, E. M., Trowbridge, J. C., Maughan, A. E., Gorman,
M. J., Brown, C. M., Prisk, T. R., Neilson, J. R. 2019. Dynamical Phase
Transitions and Cation Orientation-Dependent Photoconductivity in
CH(NH2)2PbBr3. ACS Mater. Lett. 2019, 1 (2), 260–264. Copyright 2019 American
Chemical Society.

Formamidinium lead halide perovskites typically have unusual phase behavior
compared to methylammonium and cesium analogues, including reentrant phase
transitions [116], path dependent phase transitions [104], and
crystallographically unresolvable phase transitions [92]. Similar to
methylammonium perovskites, formamidinium lead halide perovskites exhibit
dynamic degrees of freedom within the organic sublattice that freeze out with
temperature, which is reflected in the temperature dependence of the
dielectric permittivity [116, 91, 113, 92]. Anomalies in the temperature-
dependent permittivity also occur at phase transition temperatures. Despite
these similarities with methylammonium perovskites, the distinct molecular
shape and delocalized cationic $\pi$-system yield several distinctions in
formamidinium perovskites.

Formamidinium lead bromide is notable for having five temperature-dependent
anomalies in the MSDs that correlate with anomalies in the heat capacity and
in the photoconductivity (Figure 6 [92]), as well as the thermal expansion
from dilatometry [117]. Two of these phase transitions correspond to changes
in crystallographic symmetry with detectable changes in the octahedral tilting
and rotation patterns [118, 92], while three phase transitions release entropy
they do not correlate with a readily detectable change in crystallographic
symmetry [92, 94, 117]. Instead, they are characterized by changes in the
dynamic degrees of freedom of formamidinium. At the time of this writing,
these crystallographically-unresolvable phase transitions are unique among
hybrid perovskites, although formamidinium lead iodide shows unusual, sample
history dependent phase transitions that are either reentrant and/or entropy
stabilized [104, 116].

Based on the molecular geometry, the natural rotations for formamidinium are
180∘ rotations with the axis of rotation parallel to the carbon-hydrogen bond
or about the axis connecting the two nitrogen atoms, as well as a 120∘
rotation orthogonal to the molecular plane (Figure 1(a)). The point group
symmetry of the ideal lattice site occupied by formamidinium suggests that the
molecule could undergo reorientational $C_{4}$ rotations. Molecular dynamics
simulations and density functional theory calculations predict that $C_{2}$
rotations, where the axis of rotation connects the two nitrogen atoms rather
than a higher symmetry location, are the lowest energy molecular dynamics
[119, 91]; these simulations also suggest the presence of a slow reorientation
and a fast reorientation, in analogy to those observed in methylammonium
perovskites [119]. However, direct modeling of the experimental EISF is
complicated by the complex orientational potential energy landscape of
formamidinium within the nearly cuboctahedral void. The EISF resembles a
motion between idealized $C_{3}$ (120∘) and $C_{4}$ (90∘) rotations at
elevated temperatures but 180∘ rotation at lower temperature (Figure 6(e)),
consistent with changes in the relaxation times marked by the reduction in
QENS peak width across phase transitions [92].

The complex behavior of formamidinium dynamics appears to be related to its
distinct distribution of electrostatic charges. Formamidinium has a weaker
dipole moment and stronger quadrupole moment than methylammonium [120]. A pair
of molecular quadrupoles embedded in a dielectric crystal prefer to orient
perpendicular to each other in a “T” shape [121], which can couple strongly to
compressive and expansive strains from the surrounding anionic lattice [94].
This pairwise “T” shape cannot tile any Bravais lattice, and therefore the
molecular orientations of formamidinium are frustrated [92]. Frustrated
orientations are consistent with NMR results that show that formamidinium
reorients more quickly than methylammonium and with dielectric susceptibility
measurements that indicate a series of complex changes in formamidinium
dynamics at low temperatures [122, 118]. This geometric frustration
participates in the orientational phase transitions of formamidinium lead
bromide observed with QENS and could explain the path dependent phase
transitions of formamidinium lead iodide. 79Br nuclear quadrupolar resonance
(NQR) data suggest that the reorientations of formamidinium couple to changes
in the local environment around the inorganic octahedra, which aligns with
first principle calculations that show that the rotational energy barrier of
formamidinium can be tuned with substitution on the halide and metal sites
[94, 123]. Understanding the unusual phase behavior of formamidinium
perovskites is essential to understanding their properties, as each phase has
a different optoelectronic behavior [116, 92].

[] Nuclear Quadrupolar Resonance (NQR)a nuclear resonance technique performed
in the absence of a magnetic field on solids containing quadrupolar nuclei,
wherein the nuclear spin levels are split by the local chemical environment
and interrogated with radio waves.

####  5.1.3 Lead-free Hybrid Perovskite Halides

While more difficult to experimentally study (e.g., due to air sensitivity),
tin and germanium perovskites offer new opportunities for functional
properties, as they show an increased contribution of the $ns^{2}$ lone pair
electrons and lack the toxicity of lead [124, 125, 56]. The increased
stereochemical activity in tin and germaninum perovskites relative to lead
perovskites [106, 126] is expected to couple to the dynamic degrees of freedom
of methylammonium [57], although studies so far have focused predominantly on
ionic conductivity [127, 128]. However, density functional theory calculations
also indicate that tin and germanium perovskites have higher energy barriers
for methylammonium reorientations, attributed to a decrease in lattice
polarizability and compression of the $A$-site void [97, 129, 130].
Furthermore, the overall phase behavior appears to be more sensitive to the
molecular dynamics, as isotope enrichment can completely change (e.g.,
CH3NH3SnBr3 [131] vs CD3ND3SnBr3 [55]) or suppress phase transitions (e.g.,
CH3NH3GeCl3 vs CD3ND3GeCl3 [128]). Experimental validation of these
computational predictions and investigation into lone-pair coupling to
methylammonium dynamic degrees of freedom provide exciting avenues for
continued research.

###  5.2 Effects of Chemical Substitution

Most of the technologically-relevant hybrid halide perovskites for
photovoltaic applications are heavily chemically substituted (i.e., alloyed),
as exemplified by a high-performing composition:
(CH(NH2)2)_0.79(CH3NH3)_0.16Cs_0.05Pb(I_0.84Br_0.17)3 [88]. Chemical
substitution is easily accomplished synthetically in the solution-phase or
solid-state routes to these materials [132, 88], which allows for tuning of
optoelectronic properties such as the band gap and polaron mobility [133, 26].
Furthermore, the unsubstituted hybrid perovskites methylammonium lead iodide
and formamidinium lead iodide are enthalpically unstable [28] and decompose
rapidly under ambient conditions [134, 135, 136, 137]; certain substitutions
stabilize the perovskite phases [138] or slow this decomposition process [88,
139, 140]. While chemical substitution can improve phase stability, it
modifies the internal energy landscape of hybrid perovskites that influences
the atomistic dynamics and thus electron-phonon coupling [141, 26].

This section focuses on the role of cesium substitution in the two perovskite
materials described above, methylammonium lead bromide and formamidinium lead
bromides. In contrast to the hybrid materials, cesium lead bromide exhibits
orthorhombic symmetry with octahedral tilting at and below room temperature,
with the cubic and tetragonal phases accessible at temperatures above 80 ∘C
and 140 ∘C, respectively [142]. The generalized phase behavior of the
substituted alloys can be understood within the context of a pseudo-binary
phase diagram, where compositions closest to the end members behave as solid-
solutions separated by a two-phase region for intermediate compositions.
However, the detailed nature of the phase transitions requires closer
examination of the dynamic nature of the organic cations in these plastic
crystals.

####  5.2.1 Cesium Substitution in Methylammonium Lead Bromide

In methylammonium lead bromide, cesium substitution for methylammonium leads
to inhibition of the methylammonium dynamics. As discussed above,
methylammonium undergoes 90∘ rotations perpendicular and 120∘ rotations
parallel to the carbon-nitrogen bond axis [89]. Substitution of cesium for
methylammonium in the solid-solution, (CH3NH3)1-xCsxPbBr3, does not affect the
identity of these rotations; however, modeling of the QENS spectra
demonstrates that a fraction of molecules become pinned as the concentration
of cesium increases (Figure 7(a-c)) [93]. As discussed previously, the
molecular reorientations are directly responsible for a significant fraction
of the increased dielectric polarizability (Figure 5(b)).

Figure 7: Selected EISF values for (CH3NH3)1-xCsxPbBr3 extracted from QENS
spectra collected at (a) 245 K and (b) 180 K. Dashed lines are fits to the
coupled $C_{3}$ and $C_{4}$ jump model, while solid lines are fits to the
$C_{3}$ jump model. (c) The fraction of CH3NH${}_{3}^{+}$ participating in a
$C_{3}\otimes C_{4}$ rotation ($1-\alpha$) from the EISF modeling. Dashed
lines are guides to the eye. (d) MSDs as a function of temperature extracted
from QENS spectra of (CH3NH3)1-xCsxPbBr3 illustrating a smearing of
transitions with increasing $x$. The bars at the top illustrate the observed
lattice symmetry of (CH3NH3)1-xCsxPbBr3 for $x$ = 0.0, 0.1, 0.2, 0.3, and 0.4.
Reprinted with permission from Mozur, E. M. Maughan, A. E. Cheng, Y. Huq, A.
Jalarvo, N. Daemen, L. L.; Neilson, J. R. 2017. Orientational Glass Formation
in Substituted Hybrid Perovskites. Chem. Mater. 29 (23), 10168–10177.
Copyright 2017 American Chemical Society.

The pinning of molecular reorientations with cesium substitution in
(CH3NH3)1-xCsxPbBr3 correlates with a significant modification of the phase
behavior. In contrast to CH3NH3PbBr3 (Figure 4), the temperature dependent
MSDs vary smoothly with temperature and do not show abrupt changes near
crystallographic phase transition temperatures in the substituted materials
(Figure 7) [93]. With increasing substitution, crystallography indicates that
the phase transitions occur over a larger temperature range and are dependent
on temperature ramp rate [93]. The sluggish phase transitions and inhibited
dynamics indicate that methylammonium lead bromide becomes an orientational
glass upon cesium substitution characterized by disordered and frozen
methylammonium orientations, similar to bromide substitution in potassium
cyanide [58].

In analogy to spin glasses, the formation of an orientational glass relates to
frustration of order parameters related to the orientation of the molecule
[58]. One proposed source of the increased frustration on substitution is that
cesium and methylammonium prefer different coordination from the octahedral
framework. Most halide perovskites crystallize in the tetragonal $P4/mbm$
phase, characterized by in-phase octahedral rotations, as this configuration
maximizes the $A$-site coordination [143]. Hydrogen-bonding interactions
between methylammonium and the octahedral framework create a preference of the
tetragonal $I4/mcm$ phase for methylammonium perovskites, characterized by
out-of-phase octahedral rotations [144, 145, 146]. These interactions,
combined with the contraction of the $A$-site with cesium substitution,
compete and lead to the inhibition of methylammonium reorientations and the
formation of an orientational glass. It appears that substitution increases
the influence of hydrogen bonding (cf., Figure 3). Recent first principles
calculations suggest that reductions in symmetry, such as those from frozen or
inhibited methylammonium reorientations, change the binding energy of polarons
[26] or tune the magnitude of Rashba splitting [141], which suggests that
substitution impacts electronic properties through the organic cation
dynamics.

####  5.2.2 Cesium Substitution in Formamidinium Lead Bromide

The orientational phase transitions of formamidinium lead bromide are
sensitive to chemical substitution. The five phase transitions of
formamidinium lead bromide have dynamic signatures that manifest in ^1H NMR,
^14N NMR, and MSDs between room temperature and 100 K (Figure 6). However,
formamidinium lead bromide substituted with as little as 5-10% cesium exhibits
only one anomaly in ^1H NMR, ^14N NMR, or in MSDs between room temperature and
100 K, which corresponds to the high-temperature cubic to tetragonal phase
transition (Figure 8(a)) [94].

Cesium substitution effectively suppresses the four low temperature phase
transitions by interrupting the concerted changes in formamidinium dynamics.
At room temperature, ^14N NMR spectra of (CH(NH2)2)1-xCsxPbBr3 show retained
reorientational dynamics, but with an increased anisotropy of the ^14N
environment with $x$; this accompanies a clear distortion of the local ^79Br
environments, as determined from NQR spectroscopy (Figure 8(b)). The interplay
between substitution and formamidinium orientation reflects a distinct
behavior from methylammonium (Figure 3), which can be linked to the large
electrostatic quadrupolar moment of formamidinium, as depicted in Figure 8(c).
The relative orientations of formamidinium within the inorganic framework are
mediated by the electrostatic quadrupolar field, which is very sensitive to
the surrounding electric field gradient. When this interaction is mapped onto
an elastic dipole tensor, it is possible to see how a stable long-range
ordered configuration becomes challenging to produce (i.e., it is frustrated).
Introduction of compressive microstrain from the smaller cesium cation imparts
a significant change to the potential energy landscape with consequences to
the long range order and overarching phase transitions [94]. While the long-
range order is strongly influenced by the substitution, the local dynamics of
formamidinium appear to be mostly retained [94], which suggest that
formamidinium-based perovskites serve as a better host for substituted alloys
if one wishes to maintain a larger dielectric response.

Figure 8:  (a) MSDs extracted from fixed window elastic neutron scattering
spectra for (CH(NH2)2)1-xCsxPbBr3, where $x$ = 0.1 and 0.2 illustrating a lack
of discontinuities in substituted samples, in contrast to Figure 6(c). (b)
Spin-lattice relaxation time ($T_{1}$) determined from 1H NMR spectroscopy of
(CH(NH2)2)0.95Cs0.05PbBr3 illustrating one phase transition at $T\sim$ 260 K.
(c) Room-temperature ^14N NMR spectra of (CH(NH2)2)1-xCsxPbBr3 where $x$= 0 to
0.35 illustrating an increased anisotropy of cation rotations with $x$. (d)
Room-temperature ^79Br NQR spectra collected on (CH(NH2)2)1-xCsxPbBr3 showing
how increasing the level of substitution leads to an effectively continuous
distribution of possible local environments. (e) Cartoon illustrating
electrostatic potential field lines emanating from the quadrupolar moment of
formamidinium (orange density denotes negative; blue is positive) yield
attractive or repulsive interactions to the anionic framework, as depicted in
(i) and (ii). (iii) The resulting expansive and compressive strain fields map
onto an elastic dipole tensor (large arrows). (iv) In a plane, the elastic
dipoles can order; however, a favorable 3D configuration cannot be tiled. (v)
The smaller Cs+ cation introduces compressive local strain, thus causing a
preferred orientation of quadrupolar cations around the Cs+ and frustrated
orientations of the next-nearest neighboring quadrupoles. Adapted with
permission from Mozur, E. M. Hope, M. Trowbridge, J. C. Halat, D. M. Daemen,
L. L. Grey, C. P. Neilson, J. R. 2020. Cesium Substitution Disrupts Concerted
Cation Dynamics in Formamidinium Hybrid Perovskites.Chem. Mater. 32, (14)
6266–6277 Copyright 2020 American Chemical Society.

####  5.2.3 Complex, Multiple Substitutions in Hybrid Perovskite Halides

The ability to use a cocktail of cations and anions to create highly stable,
functional photovoltaic devices reflects a unique defect tolerance of hybrid
perovskites.[88]. While pseudo-binary solid-solutions formed by substitution
on either $A$-site (e.g., (CH(NH2)2)1-xCsxPbI3 [147]) or the halogen site
(e.g., CH3NH3Pb(I_$1-x$Br_$x$)3 [85] ) are prone to phase separation and
decomposition, complex, multiple substitutions on both $A$ and halogen sites
prevents this phase separation [148]. The chemical pressure from cesium
substitution for methylammonium in (CH3NH3)_0.5Cs_0.5Pb(Br_0.5I_0.5)3 appears
to have the same influence as exerting 0.2 GPa of external pressure on
CH3NH3Pb(Br_0.5I_0.5)3 and thermodynamically stabilizes the solid solution,
highlighting the importance of an “often-overlooked” $P\Delta V$ term in the
thermodynamic potential [149].

Therefore, one expects complex substitutions to also influence the organic
molecular dynamics. The inhibition with cesium substitution in methylammonium
perovskites resembles methylammonium-substituted formamidinium lead iodide, in
which lineshape analysis of ^14N NMR spectra reveals that methylammonium and
formamidinium reorientations become increasingly anisotropic with increased
substitution [122]. Substitution at the halide site, as used to increase
stability and tune the electronic bandgap [150, 88], also leads to inhibited
cation dynamics. Pump-probe two-dimensional infrared spectroscopy studies of
CH3NH3Pb(Cl_$x$Br_$1-x$)3 and CH3NH3Pb(Br_$x$I_$1-x$)3 show that both series
exhibit slower methylammonium dynamics than either end member [95]. This
supports the observation of decreased orientational disorder in methylammonium
lead bromide after chlorine substitution, as determined by Fourier difference
maps from Rietveld refinement of neutron powder diffraction [151]. Many of the
top-performing photovoltaic devices substitute a formamidinium-based host [1].
Given that the local formamidinium dynamics are not significantly inhibited by
substitution [94], any of the advantages provided by having a dynamic organic
cation are still active in formamidinium-based materials.

##  6 CONCLUSIONS AND OUTLOOK

Collective and local dynamics in hybrid perovskites underlie the
optoelectronic properties that make them of interest for photovoltaics,
radiation detection, and light emitting diodes. Therefore, characterization of
these dynamics is essential for building useful theories to describe their
behavior, especially for local dynamics that do not yet have rigorous models.
Neutron scattering plays a large role in uncovering these dynamics – the
dynamic structure factor, $S(Q,\omega)$, informs researchers of the
correlations of these molecular dynamics in both space and in time.
Complementary methods, such as NMR, dielectric spectroscopies, X-ray
scattering, molecular dynamics simulations, and DFT calculations, provide a
complete perspective across many environmental conditions. Neutron scattering
has been used to track the changes in dynamics as a function of temperature,
to describe the symmetry of molecular reorientations, and to characterize
their activation energies. These investigations demonstrate that these local
dynamics are affected by cation identity, electrostatic and hydrogen-bonding
interactions with the extended octahedral framework, and substitution. This
creates a deeper understanding of the phase behavior of hybrid halide
perovskites. Investigations into 2D perovskite materials have only just begun,
and many questions remain about the dynamic degrees of freedom of bulky
organic cations and their relationship to optoelectronic properties. Similar
questions remain for other hybrid perovskite derivatives, including zero
dimensional vacancy ordered perovskites. These insights are the basis of
rigorous structure-property relationships that will enable the rational
innovation with next-generation semiconductors.

[SUMMARY POINTS]

  1. 1.

Hybrid perovskite semiconductors are plastic crystals with dynamically
reorienting organic cations.

  2. 2.

Organic cation dynamics have a direct influence on the dielectric response of
hybrid perovskite semiconductors.

  3. 3.

Molecular reorientational dynamics exert indirect influences on the optical
and electronic properties of hybrid perovskites by way of locally broken
symmetry, electron-phonon coupling, and defect creation and motion.

  4. 4.

Chemical substitution of hybrid perovskite semiconductors influences the
thermodynamic stability and organic cation dynamics.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, under Award SC0016083. J.R.N. and E.M.M also
acknowledge support from Research Corporation for Science Advancement through
a Cottrell Scholar Award and the A.P. Sloan Foundation for assistance provided
from a Sloan Research Fellowship. The authors also thank K. A. Ross and A. E.
Maughan for insightful discussions.

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