Common acoustic phonon lifetimes in inorganic and hybrid lead halide perovskites
M. Songvilay, N. Giles-Donovan, M. Bari, Z.-G. Ye, J. L. Minns, M. A. Green, Guangyong Xu, P. M. Gehring, K. Schmalzl, W. D. Ratcliff, C. M. Brown, D. Chernyshov, W. van Beek, S. Cochran, C. Stock
CCommon acoustic phonon lifetimes in inorganic and hybrid lead halide perovskites
M. Songvilay, N. Giles-Donovan, M. Bari, Z.-G. Ye, J. L. Minns, M. A. Green, Guangyong Xu, P. M.Gehring, K. Schmalzl, W. D. Ratcliff, C. M. Brown, D. Chernyshov, W. van Beek, S. Cochran, and C. Stock School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UK Medical and Industrial Ultrasonics, School of Engineering, University of Glasgow G128QQ, UK Department of Chemistry and 4D LABS, Simon Fraser University, Burnaby, British Columbia, V5A1S6 Canada School of Physical Sciences, Ingram Building, University of Kent, Canterbury, Kent CT2 7NH, UK NIST Center for Neutron Research, National Institute of Standards and Technology,100 Bureau Drive, Gaithersburg, Maryland, 20899, USA Forschungszentrum J¨ulich GmbH, J¨ulich Centre for Neutron Science at ILL, 71 avenue des Martyrs, 38000 Grenoble, France Swiss-Norwegian Beam Lines, European Synchrotron Radiation Facility,Polygone Scientifique Louis N´eel, 6 rue Jules Horowitz, 38000 Grenoble, France (Dated: September 24, 2019)The acoustic phonons in the organic-inorganic lead halide perovskites have been reported to haveanomalously short lifetimes over a large part of the Brillouin zone. The resulting shortened meanfree paths of the phonons have been implicated as the origin of the low thermal conductivity. Weapply neutron spectroscopy to show that the same acoustic phonon energy linewidth broadening(corresponding to shortened lifetimes) occurs in the fully inorganic CsPbBr by comparing the resultson the organic-inorganic CH NH PbCl (Ref. 1). We investigate the critical dynamics near thethree zone boundaries of the cubic P m m Brillouin zone of CsPbBr and find energy and momentumbroadened dynamics at momentum points where the Cs-site ( A -site) motions contribute to the crosssection. Neutron diffraction is used to confirm that both the Cs and Br sites have unusually largethermal displacements with an anisotropy that mirrors the low temperature structural distortions.The presence of an organic molecule is not necessary to disrupt the low-energy acoustic phononsat momentum transfers located away from the zone center in the lead halide perovskites and suchdamping may be driven by the large displacements or possibly disorder on the A site. I. INTRODUCTION
The lead halide perovskites (with chemical formula A Pb(Cl,Br,I) ) display exceptional structural, optical,electronic, and charge transport properties [2–7]. In par-ticular, the semiconducting properties of CsPbBr havemade this system a promising candidate for detector ap-plications [4]. Recent theoretical and experimental workshave been focused on hybrid organic-inorganic lead halideperovskites as potentially efficient photovoltaic materi-als [8–10]. These materials combine two sublattices,formed by an inorganic lead halide framework coupledto a molecular cation (based on the A site) through hy-drogen bonding. As this coupling has been identified askey for understanding their improved efficiency in solarpower conversion, a significant number of studies havebeen devoted to investigating the coupling between themolecular and inorganic frameworks [1, 11–13] with somestudies implicating the organic molecule as the origin forthe improved thermal properties [14].Although this new class of materials has generated aconsiderable amount of theoretical and experimental in-terest, it has been shown that all-inorganic lead halideperovskites also exhibit similar efficiencies and proper-ties for photovoltaic devices [15–19]. Therefore, an openquestion is whether the organic A-site cation is essentialto the enhanced photo-electronic properties [20]. From amore fundamental point of view, the role of the cationicnature and its influence on lattice dynamics is still un-clear and needs to be further examined particularly given the suggestions that the low-energy acoustic phonons areinfluential to the electronic properties [21]. In this con-text, we investigate the low-energy acoustic phonons inCsPbBr with the goal of establishing which lattice dy-namical features are unique to the organic-inorganic vari-ants. With Cs occupying the A -site, CsPbBr is one ofthe all inorganic lead-halide perovskites.The study of cesium lead halide compounds started 40years ago when the first structural characterization wasperformed using neutron inelastic scattering in CsPbCl by Fujii et al. , who reported the temperature dependenceof superlattice Bragg peaks through the structural transi-tions, as well as the dispersion of acoustic phonons acrossthe Brillouin zone [22]. In an attempt to characterizethe structural transitions in this compound, this studyshowed the presence of overdamped phonons at the M and R zone boundaries, which prevents the observationof any phonon softening near the transitions. A detailedstudy of the structural transitions was later performed byHua et al. [23] who reported three structural transitionsin CsPbCl : a cubic to tetragonal transition, followed bya tetragonal to orthorhombic phase transition and finallyan orthorhombic-orthorhombic transition.A neutron diffraction study was also performed on thebromine compound [2] which exhibits only two struc-tural transitions: one first order transition from a cubicphase described by the space group P m m to a tetrag-onal symmetry around 400 K (space group P /mbm ),and a second order transition to an orthorhombic phasearound 360 K (space group P mbn ). The structural tran- a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p C u b i c (a) (b) O r t h o r h o m b i c T e t r a go n a l X Γ M R
300 320 340 360 380 400 420Temperature (K)020406080100 I n t en s i t y ( a r b . un i t s ) R pointM pointX point
FIG. 1. ( a ) Crystallographic structure of CsPbBr in thecubic phase. The Pb site is represented with the gray sphere,the Br ions are in light brown and the Cs ions are in blue. Thefigure on the top shows the high symmetry points in the firstBrillouin zone of a primitive cubic lattice. X , M and R rep-resent Q = ( , , , ( , , , , ) respectively. ( b )Temperature dependence of the superlattice intensities at thethree zone boundaries X (2, ,0), M ( , ,0) and R ( , , ).The colored dashed lines indicate the two transition temper-atures. sitions in both chlorine and bromine compounds werethen further investigated with ultrasound measurementsthat reported several anomalies in the longitudinal andtransverse sound velocities at the structural transitions,indicative of changes in the elastic coefficients [24, 25].Since then, many experimental and theoretical studieshave been devoted to similar compounds, such as cesiumtin halides [26–28].This paper reports a neutron scattering study of thetransverse acoustic phonons in the all-inorganic per-ovskite CsPbBr towards the three high symmetry pointsof the cubic P m m Brillouin zone Q X = (2 , , Q M = ( , , Q R = ( , , ). We show the ex-istence of both critical fluctuations and resolution lim-ited Bragg peaks at specific zone boundaries when ap-proaching the orthorhombic transition, concurrently withthe presence of damped phonon modes indicating short-ened lifetimes. Based on powder diffraction data show-ing strong anisotropic Cs and Br displacements in thetetragonal and orthorhombic phases, we speculate thatthe strong phonon damping originates from anharmoniceffects related to this anisotropy. Based on these re-sults, we discuss the importance of displacive transi-tions in this compound along with the influence of possi-ble anisotropic A -site fluctuations affecting the acousticphonon lifetime. This work therefore highlights similarmechanisms affecting the harmonic phonons as in the hy-brid counterparts, which may explain their comparableefficiencies in terms of thermal and opto-electronic prop-erties. This also illustrates that the strong phonon damp-ing is possibly universal across the lead-halide perovskitesand does not require the presence of an organic moleculeon the A site. II. EXPERIMENTAL DETAILS
Sample Preparation:
Single crystals of CsPbBr wereprepared using the Bridgman method. First, a powder ofCsPbBr was synthesized by solid state reaction follow-ing the procedure described in [4]. A bright orange in-got resulted from the reaction and was ground to obtaina homogeneous powder of CsPbBr . The powder wasthen pelleted and sealed in an evacuated quartz ampoulewhich was placed into a 3-zone horizontal tube furnace.The hottest zone was set to 600 ◦ C and a temperaturegradient of 10 ◦ C/cm was applied. Orange transparentcrystals were obtained and checked with neutron Lauediffraction using OrientExpress (ILL, Grenoble).
Neutron powder diffraction measurements:
Powderneutron diffraction measurements were performed on theBT1 diffractometer (NIST, Gaithersburg) using Cu(311)( λ = 1.5399 ˚A) and Ge(311) ( λ = 2.079 ˚A) monochro-mators for the orthorhombic and tetragonal phases, re-spectively. Rietan [29] was used to perform Rietveldrefinement of powder neutron diffraction data. Variabletemperature synchrotron X-ray powder diffraction datawere collected at the Swiss-Norwegian beamline BM01(SNBL) at the ESRF (France) with an incident wave-length of λ = 0.956910 ˚A. The Rietveld refinements ofthis synchrotron X-ray powder diffraction data were per-formed using the Fullprof suite [30].
Neutron inelastic measurements:
Neutron inelasticspectroscopy was performed on the thermal triple-axisspectrometer IN22 (ILL, Grenoble) with constant k f =2.662 ˚A − , using a PG filter between the sample andthe analyzer to remove higher order contamination. Fur-ther measurements as a function of temperature wereperformed on the thermal triple-axis instrument BT4(NIST, Gaithersburg) with constant k f = 2.662 ˚A − us-ing PG filters between the monochromator and the sam-ple, and between the sample and the analyzer. Higherresolution measurements were also carried out on thecold triple-axis spectrometer SPINS (NIST, Gaithers-burg) with k f = 1.55 ˚A − using a Be filter between thesample and the analyzer.All phonon measurements were performed in the(H K 0) and (H H L) scattering planes. As super-latticereflections are expected at the X , M and R zone bound-aries, measurements were performed towards specific di-rections where the structure factor was non-zero, as de-scribed in [22]. Throughout the paper we use the zoneboundary notation taken with respect to the cubic unitcell shown in Fig. 1 ( a ). Diffraction measurements in-vestigating the three zone boundaries were done both inthe (H K 0) and (H H L) planes. With the 0.5 g samplealigned in the (H K 0) plane, the phonon dispersions ofthe TA and TA modes could be extracted and anothersample aligned in the (H H L) plane was used to mea-sure the phonons toward the Q R = ( , , ) zone bound-ary. The TA mode corresponds to the acoustic phononspropagating along the [1 0 0] direction with a polariza-tion along [0 1 0] or [0 0 1]. Using the equations of motionoutlined in Ref. 31, the velocity of this phonon can berelated to the C elastic constant. The TA phononmode propagates along [1 1 0] with a polarization along[1 1 0], and the lim q → slope of the dispersion depends on( C − C )/2. Given that the neutron cross section forphonon scattering scales as ( (cid:126)Q · (cid:126)e ) , where (cid:126)e is the phononeigenvector, transverse scans near (cid:126)Q = (2 0 0) ± (0 q phonon while scansnear (cid:126)Q = (2 2 0) ± ( − q q
0) give the TA mode. III. PHONON MOMENTUM DEPENDENCE -FREQUENCY AND LIFETIMES
Acoustic phonons were mapped out in the (H K 0)scattering plane and we report in this section anomaliesfound in the phonon dispersion at particular points of theBrillouin zone. The phonon dispersions were measuredin both the orthorhombic and cubic phases at 300 K and420 K respectively, around the (2 0 0) and (2 2 0) Braggpositions in the [2 q
0] and [2- q q
0] directions. Thesecorrespond to the directions from the Γ zone center pointtowards the X and M Brillouin zone boundary symmetrypoints, as shown in Fig. 1( a ). Dispersions in both direc-tions were measured on the thermal triple-axis spectrom-eter IN22 (ILL) and were extended to low momentumtransfer, to approach the q → q
0] direction.The [2- q q
0] direction is not reachable with cold neu-trons due to kinematic constraints of neutron scattering.Figures 2 ( a ) and 3 ( a ) show constant-Q scans in the[2 q
0] and [2- q q
0] directions, respectively, at T =300 K, measured on IN22 with thermal neutrons. Har-monic phonon modes can be observed and show a depen-dence in momentum transfer. The phonon energy posi-tion ω could be extracted as a function of q by fittingthe experimental data using a damped harmonic oscilla-tor model and a constant background: S ( (cid:126)Q, ω ) = ... [1 + n ( ω )] I (cid:18) γ γ + ( ω − ω ) − γ γ + ( ω + ω ) (cid:19) , where [1 + n ( ω )] is the Bose factor, I is a constant and γ is the phonon energy linewidth related to the phononlifetime via γ ∼ τ . The above expression of the neu-tron scattering cross section takes into account both neu-tron energy gain and loss and obeys detailed balance [32].The elastic scattering was fitted to a Gaussian centeredaround E=0 with a width fixed to the instrument reso-lution extracted from vanadium incoherent scattering.Fig. 2 ( b ) and 3 ( b ) show the resulting dispersions fromthe zone center Γ to the X zone boundary (TA mode)and Γ to M (TA mode), respectively. Near the (2 0 0)and (2 2 0) zone centers, the phonons were found to bewell defined in energy and momentum. A direct compar-ison with the phonon dispersion in the organic-inorganic (a) (b)(c) TA T = 300 K
E (meV) I n t e n s i t y ( a r b . un i t s ) q = 0.20q = 0.225q = 0.25q = 0.275q = 0.30q = 0.325q = 0.35q = 0.40012345 E ( m e V ) SPINS 300KIN22 300KSPINS 420KMAPbCl (2 q 0) (r.l.u) P honon li n e w i d t h ( m e V ) SPINS 300KIN22 300KSPINS 420KMAPbCl FIG. 2. ( a ) Constant-Q cuts through the acoustic phononTA for several Q positions from the (2 0 0) Bragg peak to-wards the X point. ( b ) Dispersion curves associated to theTA mode towards the X point. The data is compared tothe organic-inorganic compound CH NH PbCl in its cubicphase (gray circles, from [1]). The blue and grey dashed linesare a fit to a sine function performed near the q → c )Q-dependence of the TA phonon linewidths extracted fromthe constant-Q cuts as described in the text and comparedto the phonon linewidth for CH NH PbCl (grey diamonds,from [1]). The grey dashed areas represent the instrumentalresolution of SPINS ( q < q > counterpart CH NH PbCl in each direction (taken from[1]), measured on the IN22 spectrometer and extractedfollowing the same fitting procedure, is shown in grey.At low momentum transfer, near the q → C from the TA modeand ( C − C ) / mode. Table I sum-marizes the extracted elastic constant values along witha comparison with the chlorine counterpart and the hy-brid compound CH NH PbCl . The elastic constants (a) (b)(c) TA T = 300 K
E (meV) I n t e n s i t y ( a r b . un i t s ) q = 0.125q = 0.13q = 0.150q = 0.175q = 0.200q = 0.250q = 0.300q = 0.35001234567 E ( m e V ) IN22 300KIN22 420KMAPbCl (2-q 2+q 0) (r.l.u) P honon li n e w i d t h ( m e V ) IN22 300KIN22 420KMAPbCl FIG. 3. ( a ) Constant-Q cuts through the acoustic phononTA for several Q positions from the (2 2 0) Bragg peak to-wards the M point. ( b ) Dispersion curves associated with theTA mode towards the M point. The data is compared tothe organic-inorganic compound CH NH PbCl in its cubicphase (grey triangles, from [1]). The blue and grey dashedlines are a fit to a sine function performed near the q → c ) Q-dependence of the TA phonon linewidths ex-tracted from the constant-Q cuts as described in the text andcompared to the phonon linewidth for CH NH PbCl (greysquares, from [1]). The blue and grey dashed lines are a guideto the eye. The grey dashed area represents the instrumentalresolution of IN22. for our Br variant are generally lower than the Cl andorganic-inorganic compounds. Given the larger size ofBr, the decrease may be attributed to an increase in lat-tice constants. It can be noted that no clear change inthe dispersion curves occur between 300 K and 420 K onthe energy scale ( ∼ THz) probed with neutron scatter-ing. The temperature dependence of the lattice dynamicswill be discussed later. Moreover, the TA mode becomessignificantly flat, dispersing little in energy towards the M zone boundary.While acoustic phonons are well defined near the zonecenter with γ < ω , they become much broader in en-ergy towards the zone boundaries, indicative of a short-ened lifetime. The phonon linewidth was extracted asa function of q and shows a shorter lifetime for smallerwavelength excitations, as shown in Figures 2 ( c ) and 3( c ). The effect is the most dramatic for the TA phononmode, towards the M point as phonons could not be ob-served above q = 0.35. This feature was also reportedin CsPbCl [22]. As shown in grey, the momentum de-pendence of acoustic linewidth is also very similar tothe organic-inorganic hybrid perovskite CH NH PbCl [1] near the zone boundary. At the momentum transfersnear the zone center, the linewidth enters the resolutionof the spectrometer and higher resolution probes are re-quired to investigate the phonon lifetime in this region ofmomentum. IV. TEMPERATURE DEPENDENCEA. Static properties
A temperature dependence study was carried outto characterize the different structural transitions inCsPbBr . The intensity of the elastic superlattice re-flections at the X point (2, ,0), M point ( , ,0) and R point ( , , ) was measured as a function of tempera-ture on the IN22, BT4 and SPINS spectrometers, respec-tively. New nuclear Bragg peaks appear at all three zoneboundaries, implying a doubling of the unit cell alongthe crystallographic directions on entering the tetrago-nal and orthorhombic phases from the high temperaturecubic phase. As shown in Fig. 1( b ), a sudden jump in theintensity occurs at the M point around T = 410 K whenthe unit cell transforms from cubic to tetragonal. Whenentering the low-temperature orthorhombic phase aroundT = 360 K, superlattice intensities simultaneously appearat the X and R points. While the discontinuous evolu-tion of the intensity at the M point suggests a first-ordertransition, the intensity at the X and R points seem tofollow a second-order transition, as previously reportedin [2, 23].Previous work on CsPbCl analyzing the structurefound irregular behavior on the Cs and Cl sites whichcould be modelled as either an anomalously large ther-mal parameter [33] or site disorder [34]. Following this,we have carried out an analysis of the temperature de-pendence of the thermal parameters for CsPbBr usingpowder neutron diffraction (Fig. 4) and synchrotron X-ray diffraction. The refined values as a function of tem-perature, for the components of the thermal parametersalong the three directions in space ( B , B and B )for Br and Cs are displayed in Figure 4 ( b ) and are illus-trated in real space in Figure 5. The blue ellipsoids inthe left panels of Fig. 5 represent thermal displacementsof the Cs ions in the tetragonal and orthorhombic phases TABLE I. Comparison of elastic constants extracted from ultrasound measurements in the cubic phase for CsPbCl and neutroninelastic measurements in the cubic phase for CsPbBr and MAPbCl (MA = CH NH ).CsPbCl (from [25]) CsPbBr (this work) MAPbCl (from [1]) C (GPa) 5.04 2.45(3) 3.00( C − C )/2 (GPa) 8.95 4.45(5) 10.8 and the right panels show the octahedral tiltings associ-ated with both structural transitions. It should be notedthat the powder diffraction data showed a slightly differ-ent temperature for the transition from the tetragonal toorthorhombic phase (found around 365 K), compared tothe single crystal sample (360 K). Like in CsPbCl , thethermal parameters values are anomalously large for theBr and Cs sites across all temperatures.In the cubic phase, Figure 4 ( b ) (bottom panel) showsthat the Cs motions are quasi-isotropic while the Br mo-tions (top and middle) are large and anisotropic in com-parison, as also previously reported in [33] for CsPbCl .When going through the tetragonal and orthorhombictransition however, the Cs site also becomes anisotropicas indicated by the elongated shapes of the thermal el-lipsoids in Fig. 5. Moreover, as illustrated in Figure5 ( b ), the results also show an off-center rearrangementof the A-site cation in the low-temperature phase, indi-cated by the arrows. The shape and orientation of theellipsoids representing the Cs thermal displacements fol-lows the PbBr octahedra tilting in both tetragonal andorthorhombic phases, which suggests a strong couplingbetween the A-site cation and the halogen anions, as dis-cussed later. B. Lattice dynamics
The temperature evolution of the lattice dynamics wasalso investigated. As mentioned and shown in Fig. 2 and3, no clear change in the phonon dispersion curves forCsPbBr between the high-temperature cubic phase andthe low-temperature orthorhombic phase, was observednear the zone center. This contrasts with the case ofCH NH PbCl where the elastic constants, and hencethe acoustic phonon dispersion, shows a discontinuity atthe low temperature structural transitions. [1] This maybe indicative of an underlying ferroelastic character assuggested by ultrasound measurements. [35]We now discuss the temperature dependence at the X (2, ,0) and M ( , ,0) zone boundaries, through the twostructural transitions. Figures 6 ( a )-( b ) and 7 ( a ) showconstant-Q scans at the M , R , and X points respectivelyin the cubic (T = 420 K) and orthorhombic (T = 300 K)phases. While the acoustic phonons remains unobserv-able at the M and R points in both phases, the TA phonon shows a broadening at the X point when cool-ing into the orthorhombic phase. As also reported inCsPbCl [22], the presence of highly damped phononsaround the M and R points does not allow for a direct (a)(b) BT1
T = 365 K λ = 2.079 Å B Br B Br B T he r m a l pa r a m e t e r B ( Å ² ) Br B Br B Br B
300 320 340 360 380 400 420Temperature (K)0123456 Cs B Cs B Cs B (°) FIG. 4. ( a ) Powder neutron diffraction data performed onBT1 ( λ = 2.079 ˚A) at T = 365 K. The red dots and blackline represent the measured and calculated intensities, respec-tively. The blue stick marks indicate the calculated reflectionsand the blue line represents the difference between measuredand calculated intensities. ( b ) Thermal parameters for thethree spatial components associated with the Br , Br andCs sites, respectively, as a function of temperature. The col-ored areas and dashed lines indicate the tetragonal and or-thorhombic transitions. (a)(b) Tetragonal phaseOrthorhombic phase Br Br FIG. 5. ( a )-( b ) Thermal displacements of Cs ions (blueellipsoids) and PbBr octahedra tiltings (right panels) in thetetragonal and orthorhombic phases, respectively. The arrowsindicate atomic displacements and octahedral rotations. measurements of the soft modes at these zone bound-aries throughout the transitions. Fig. 7 ( b ) displays therelative change in linewidth γ and energy position ω ofthe TA mode, starting from the high temperature phase(set as the reference) and cooling down towards 300 K.With decreasing temperature, the TA mode seems tosoften very slightly while there is a clear increase in thephonon linewidth when entering the orthorhombic phase,indicating a shortening of the phonon lifetime.We now discuss the relaxational scattering aroundE = 0 meV for the three zone boundaries X (2, ,0), M ( , ,0) and R ( , , ). Energy scans were carried outas a function of temperature, using the cold triple-axisspectrometer SPINS for the measurements at the R and X points (Fig. 8 ( a ) and 9 ( a ) respectively) and the ther-mal triple-axis spectrometer IN22 for the measurementsat the M point (Fig. 9 ( b )). At the X and M zoneboundaries, the data shows a rapid increase in the inten-sity when decreasing the temperature, while the widthin energy remains resolution limited. It should be notedthat the low temperature data corresponds to the ap-pearance of new Bragg peaks, with resolution tied to theinstrumental Bragg resolution which is much narrowerthan the resolution extracted from the incoherent sig-nal of vanadium as shown with the black dashed line. Incontrast, at the R point, critical dynamic fluctuations be-yond the instrumental resolution could be observed abovethe tetragonal-to-orthorhombic phase transition.In order to parameterize the observed neutron scat- E (meV) I n t e n s i t y ( a r b . un i t s ) T = 300KT = 420K0 1 2 3 4 5
E (meV) I n t e n s i t y ( a r b . un i t s ) T = 300KT = 420K
M point
IN22 (a)(b)
R point
SPINS
FIG. 6. ( a ) Constant-Q cuts through the TA mode atthe M point ( , ,0) measured on IN22 at 300 K and 420 K.( b ) Constant-Q cuts through the TA mode at the R point( , , ) measured on SPINS at 300 K and 420 K. tering cross section, the experimental data was fitted tothe sum of a Gaussian with width fixed to the experi-mental resolution derived from the incoherent scatteringfrom vanadium (shown with black dashed lines) and asecond narrower Gaussian function. Fig. 8 ( c ) shows thetemperature evolution of the extracted energy linewidth.In particular, the linewidth shows a first jump when de-creasing temperature from the cubic to the tetragonalphase. Then, when the system enters the orthorhombicphase, the linewidth further decreases and reaches a min-imum value corresponding to the Bragg resolution whilethe intensity increases. The presence of critical fluctua-tions in energy is concomitant with a narrowing in mo-mentum as observed with elastic transverse scans shownin Fig. 8 ( b ) when decreasing temperature. The elasticscattering, which is broad in momentum above the cubicphase, progressively sharpens along with an increase inintensity, indicating a critical divergence of the dynamiccorrelation length. The temperature dependence of thecorrelation length ξ was extracted by fitting the elasticdata with a Lorentzian of the form: I ( q ) = I γ ( γ + q ) , (1) E (meV) I n t e n s i t y ( a r b . un i t s ) T = 300KT = 420K300 320 340 360 380 400 420
Temperature (K) p / p ( T = K ) (a) (b) X point
FIG. 7. ( a ) Constant-Q cuts through the TA mode at the X point measured on IN22 at 300 K (orthorhombic phase) and420 K (cubic phase). ( b ) Temperature dependence of the en-ergy position (black circles) and the energy linewidth (whitecircles) of the TA phonon at the X (2, ,0) zone boundary.The vertical dashed line indicates the orthorhombic transi-tion. where I is a constant, γ = 1 /ξ is the inverse of the real-space correlation length and q is the length of the wave-vector measured from the ( , , ) position. The spatialcorrelation length, plotted as a function of temperaturein Fig. 8 ( d ), shows a divergence when going towards theorthorhombic transition at T = 360 K, as indicated bythe dashed line. A power-law fit of the correlation length,following a mean-field analysis, as a function of tempera-ture using a form ξ ( T ) = ξ (cid:0) T − T cT c (cid:1) ν was also performed(shown with the red line) and gave an exponent of ν =-0.53 ± − predicted frommean field theory [36]. V. DISCUSSIONA. Energy broadening of the TA mode We first address the significant energy broadening ofthe phonons towards the M zone boundary of the TA phonon mode. Temporally broad acoustic phonons havealso been reported for the hybrid organic-inorganic halide -1 -0.5 0 0.5 1 E (meV) I n t e n s i t y ( a r b . un i t s ) (0.5 0.5 L) (r.l.u) I n t e n s i t y ( a r b . un i t s )
300 350 400
Temperature (K) L i n e w i d t h ( m e V )
350 360 370 380 390 400
Temperature (K) ( Å ) R point (a) (b)(c) (d)
FIG. 8. ( a ) Energy scans measured on SPINS at the R point ( , , ) in the cubic and orthorhombic phases. Theblack dashed line indicates the instrumental energy resolu-tion, extracted from the incoherent signal of vanadium. ( b )Elastic scans measured on SPINS at the R point ( , , ) atseveral temperatures. The black dashed line shows the Bragginstrumental resolution in momentum. ( c ) Temperature de-pendence of the energy linewidth extracted from the quasi-elastic scans at the R point. ( d ) Temperature dependence ofthe spatial correlation (inverse of the momentum linewidth)extracted from the elastic scans at the R point. The verti-cal dashed line indicates the temperature transition to theorthorhombic phase. perovskites CH NH PbCl [1] and CH NH PbI [14] andattributed to the molecular dynamics [11, 12] stronglyaffecting the acoustic phonon modes through hydrogenbonding. It was further suggested [14], with lattice dy-namics calculations, that these anharmonic effects aredue to low-energy optical phonon modes arising from theorganic cation. The resulting short mean free path of thephonon resulting from this dampening was suggested tobe the origin of the ultra-low thermal conductivity in thiscompound.Figures 2 and 3 illustrate that this same phonon damp-ing is present in CsPbBr , even without the presence ofan organic molecule. However, in the case of CsPbBr ,the powder diffraction data presented in Fig. 4 showslarge anisotropic Cs and Br motions in both the tetrago-nal and orthorhombic phases. These anisotropic Cs dis-placements respond in concert with octahedra tilting,suggestive of a strong coupling between the A cation -1 -0.5 0 0.5 1 E (meV) I n t e n s i t y ( a r b . un i t s ) -2 -1 0 1 2 E (meV) I n t e n s i t y ( a r b . un i t s ) X point
SPINSM point
IN22 (a)(b)
FIG. 9. ( a )-( b ) Energy scans measured on SPINS and IN22at the X and M points, respectively, in the cubic and or-thorhombic phase. and the halogen anions. The large anisotropic displace-ments on the A -site may, in turn, damp tilt modes ofthe PbBr octahedra manifesting as energy broadeningnear the M zone boundary. As discussed below, this isconsistent with the structure factors of neutron scatter-ing, [22, 37] and is supported by calculations [26–28] ona series of fully inorganic lead halide perovskites, whichhave reported anharmonic fluctuations, including octa-hedral tilting. In particular, Refs. [27, 38] stress the im-portance of the coupling between the Cs + displacementsand the tilt distortions in order to explain the dynami-cal instabilities in the high and intermediate temperaturephases.While the energy widths of the acoustic phonons aresimilar in CsPbBr and CH NH PbCl , there are somedifferences between the organic and inorganic perovskitevariants. Contrary to the organic compound, the TA and TA phonon linewidths in CsPbBr , near the zonecenter, do not show any measurable change in tempera-ture (on the THz scale) while the organic variant showsa broadening of the long wavelength acoustic phononsin the intermediate tetragonal phase. This additionaltemperature dependent broadening may be the result ofthe molecular motion and this indeed has been suggestedby Raman scattering (at Q → B. Temperature dependent dynamics and staticsnear the zone boundaries
The structural transitions in the lead halide perovskitecompounds CsPb(Cl,Br) were previously studied byneutron scattering [2, 22, 23] and confirmed by ourdiffraction results in Fig. 1. CsPbCl exhibits three tran-sitions while the Br counterpart undergoes only two witha cubic to tetragonal transition at ∼
400 K and an or-thorhombic unit cell forming below 360 K. Group theoryanalysis [2] predicts that in CsPbBr , the higher temper-ature transition is caused by the condensation of the M mode at the M point ( , , R point in the cubic Brillouin zonewhich transforms into the Z point of the tetragonal Bril-louin zone. We note that both R and X points of thecubic lattice are equivalent in the tetragonal phase andhence show simultaneous superlattice reflections. The R mode associated to the R point splits into a Z modeand a doubly degenerate Z mode. It is predicted thatthe transition is driven by a softening of the Z modeswhich are based on the motions of both the Cs and Brions.As discussed in Ref. [23], and in relation to neutronscattering structure factors in [37, 40], the M point in-volves the tilting of the PbCl octahedra around the[0 0 1] cubic axis (also shown in the right panel of Fig. 5( a )), while the R point (measured in the (H H L) scatter-ing plane) involves displacive distortions of the octahedraaccompanied with the displacement of Cs ions. The A -site Cs motions may explain the critical scattering (onthe THz frequency scale) observed at the Q R = ( , , )point and presented in Fig. 8 ( a )-( d ), yet simultaneouslynot measurable at the Q X = (2 , ,
0) point. While both Q R and Q X are the same zone boundary (termed Z )in the tetragonal phase, they are different momentumpoints with Q X having an identical zero structure factorfor the Cs site while the Cs A -site motion gives a nonzeroneutron structure factor at Q R . This is consistent withthe group theory analysis outlined above and the powderdiffraction data shown in Fig. 4.Our temperature dependent results are consistent withcritical fluctuations of the Cs A -site and Fig. 8 ( c ) il-lustrates two-steps suggestive of Cs dynamics in boththe high temperature cubic phase and the intermedi-ate tetragonal phase. This is supported by theoreticalwork [27] highlighting the importance of displacive A -sitemotion and its coupling to octahedra tilt modes. Con-currently, we note that dynamic critical fluctuations arenot observable in CH NH PbCl [1] at the M and X zoneboundaries, but show a sudden increase of resolution lim-ited Bragg peaks. This may indicate that the transitionsin the inorganic variants is dominated by a coupling to re-laxational dynamics rather than displacive [41–44]. Thislater point requires further study with higher resolutiontechniques.The comparison between the organic-inorganic per-ovskites and the fully inorganic variants indicate a strongcoupling between the A site and the tilt distortions, evenin the absence of hydrogen bonding. It also appears thatthe influence of the A site, either disordered or stronglyfluctuating, are ubiquitous to the lead halide perovskites.This in turn results in highly damped acoustic phononfluctuations and may explain the apparently universallow thermal conductivity found both in fully inorganic[16] and organic-inorganic lead halide perovskites [14]. VI. CONCLUSION
We report short acoustic phonon lifetimes in the fullyinorganic CsPbBr . The phonon lifetimes are similar to organic variants, indicating that the presence of an or-ganic molecule on the A -site and the resulting hydrogenbonding are not crucial for large damping to appear in alarge part of the Brillouin zone. ACKNOWLEDGMENTS
We acknowledge funding from the EPSRC, STFC andCarnegie Trust for the universities of Scotland. The workwas also supported by the Natural Sciences and Engi-neering Research Council of Canada (NSERC, Grant No.203773) and the U. S. Office of Naval Research (GrantNo. N00014-16-1-3106). We acknowledge the supportof the National Institute of Standards and Technology,U.S. Department of Commerce, in providing the neutronresearch facilities used in this work. [1] M. Songvilay, M. Bari, Z.-G. Ye, G. Xu, P. M. Gehring,W. D. Ratcliff, K. Schmalzl, F. Bourdarot, B. Roessli,and C. Stock, Phys. Rev. Mater. , 123601 (2018).[2] S. Hirotsu, J. Harada, and M. Iizumi, J. Phys. Soc. Jpn , 1393 (1974).[3] V. Babin, P. Fabeni, M. Nikl, G. P. Pazzi, I. Sildos, N. Za-zubovich, and S. Zazubovich, Chem. Phys. Lett. , 31(1999).[4] C. C. Stoumpos, C. D. Malliakas, J. A. Peters, Z. Liu,M. Sebastian, T. C. Chasapis, A. C. Wibowo, D. Y.Chung, A. J. Freeman, B. W. Wessels, and M. 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