Pseudospin-phonon pretransitional dynamics in lead halide hybrid perovskites

The low-frequency lattice vibrations and relaxations are investigated in single crystals of the four 3D hybrid organolead perovskites, ${\mathrm{MAPbBr}}_3$, ${\mathrm{FAPbBr}}_3$, ${\mathrm{MAPbI}}_3$, and $\alpha \text{−}{\mathrm{FAPbI}}_3$, at the Brillouin zone center using Raman and Brillouin scattering and at the zone boundary using inelastic neutron scattering. The temperature dependence of the ${\mathrm{PbX}}_6$ lattice modes in the four compounds can be renormalized into universal curves, highlighting a common vibrational dynamics at the cubic to tetragonal transition. In particular, no soft vibration is observed excluding a displacive-like transitional dynamics. The reorientational (pseudospin) motions of the molecular cations exhibit a seemingly order-disorder character recalling that of plastic crystals, but attributed to a secondary order parameter. At ultralow frequency, a quasielastic component evidenced by Brillouin scattering and associated to the unresolved central peak observed in neutron scattering, is attributed to center of mass anharmonic motions and rattling of the molecular cations in the perovskite cavities. Its partially unexpressed critical behavior at the transition points toward the general importance of defects in HOPs preventing the net divergence of order parameter correlations at the critical temperatures.

Figure 1

Raman spectroscopy in MAPB across the three phase transitions: cubic (C), tetragonal (T), intermediate (I), and orthorhombic (O). (a) Phonon frequency and (b) phonon linewidth (zoom in the inset) resulting from a fit with damped harmonic oscillators. (c) Low-frequency Raman responses in the tetra (T), intermediate (I), and orthorhombic phase (O). The main phonon bands ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$, are identified by red, blue, and green filled squares [panels (a),(b),(c)] and colored double arrows [panel (c)].

Figure 2

Examples of Raman spectra (symbols) of various HOPs around their respective tetragonal to cubic phase transitions with fits comprising overdamped harmonic oscillators and a central component (plain lines). For each compound, the individual spectral line shapes $S_i\left(\omega \right)$ ($i=1$ in red, $i=2$ in blue, $i=3$ in green, $i=4$ in gray) are shown for the lowest temperature spectra (bottom, tetragonal phase) and for the highest temperature spectra (top, cubic phase). For clarity, the spectra have been shifted up by a constant value.

Figure 3

Inelastic neutron scattering constant-Q scans around the Brillouin zone boundary at the M point Q = (5/2 1/2 0) (FAPB) and the R points Q = (5/2 1/2 1/2) (MAPI) and (3/2 3/2 1/2) (MAPB and FAPI). All experiments done on the thermal instrument (1T) except for FAPI measured on cold TAS (4F2). For each compound, superstructure Bragg peaks occur in the tetragonal phase at these Q positions. The lines are fits as explained in the text. The spectra at the two highest temperatures have been shifted up by a constant value of 50 and 100 counts.

Figure 4

(a) Depolarized Raman spectra ($I_{\perp }$) with incident polarization parallel to [100] and $q\parallel$ [001] of MAPB, FAPB, and MAPI in the cubic phase. The inset shows the polarized spectra ($I_{//}$) of MAPI. The dashed line indicates approximately the position of the peak maximum of ${\omega }_1$. (b) Temperature dependence of the low-frequency Raman response in MAPI (${\mathrm{I}}_{\perp }$).

Figure 5

Low-frequency INS response in FAPB at the M point Q = (3/2 1/2 0). Spectra at different temperatures fitted with the sum a Lorenztian curve given by Eq. (2) accounting for the QE, and a delta function (CP) accounting for the elastic-like central peak, convoluted by the instrument resolution function (dashed and dotted lines at 270 K, respectively). The spectra have been shifted by 200 counts for clarity. The inset shows the temperature dependence of the elastic intensity at the M and R points.

Figure 6

Brillouin spectra in FAPB (symbols) and their fits (lines): transverse acoustic (TA) and longitudinal acoustic (LA) phonons (cyan), quasielastic (green), and the total (orange), which includes in addition a constant background accounting for the broad QE component (note that 30 GHz $\sim 1$ ${\mathrm{cm}}^{-1}\sim$ 0.12 meV).

Figure 7

DHO-model: Fitting parameters of the Raman mode of lowest frequency ${\omega }_1$ in MAPB, FAPB, MAPI, and FAPI. (a) Temperature dependence of the frequencies ${\omega }_{01}$ and (b) damping ${\mathrm{\Gamma }}_1$. (c) Normalized relaxational frequencies ${\omega }_{\mathrm{Rel}}$(T)-${\omega }_{\mathrm{Rel}}(\mathrm{T}\ge {\mathrm{T}}_c$) as a function of T-${\mathrm{T}}_c$, and (d) normalized damping ${\mathrm{\Gamma }}_1$(T)-${\mathrm{\Gamma }}_1(\mathrm{T}\ge {\mathrm{T}}_c$) as a function of T-${\mathrm{T}}_c$. (e) Spectral line shapes in the cubic phase and (f) temporal response of the atomic displacements.

Figure 8

Characteristic frequencies and linewidth of the low frequency Raman mode of MAPB, FAPB, MAPI, and FAPI, in the cubic phase.

Figure 9

Analysis of the quasielastic INS response in FAPB at the M point Q = (3/2 1/2 0). (a) Area of the QE and CP components: $A_{\mathrm{QE}}\times {\gamma }_{\mathrm{QE}}$ for the former (filled squares), and the delta function amplitude for the latter in addition to the flat incoherent scattering (open squares). The line is a guide for the eyes. (b) Temperature dependence of the damping of the QE component, ${\gamma }_{\mathrm{QE}}$ fitted with an Arrhenius law (line).

Figure 10

(a) Area (full symbols) and half width at half maximum (empty symbols) of the narrow QE in FAPB, and (b) frequency of the LA phonon associated to the elastic constant ${\mathrm{C}}_{11}$ in FAPB and MAPB, as a function of temperature. The dashed lines indicate the cubic to tetragonal transition in MAPB (240 K) and FAPB (265 K).