Introduction

The vibrational properties of solids are pivotal for many physical phenomena, including but not limited to phase stability and thermal conduction. In crystalline solids, the vibrational spectrum is commonly described in terms of phonons as quasi-particle representations of the lattice vibrations. The phonon frequency ω0 is typically much larger than the damping Γ, and the phonon relaxation time τ = 2/Γ is thus much longer than the oscillation period, such that the quasi-particle picture is well motivated1,2,3,4,5,6. In this so-called underdamped limit, the relaxation time decreases as the damping Γ increases.

By comparison, there are far fewer cases when phonon modes become overdamped, i.e., ω0τ < 17,8. This can occur either due to large damping or for very soft modes, usually in the immediate vicinity of a phase transition, for example, in the case of body-centered cubic Ti9,10,11, rotationally disordered 2D materials12, in ferroelectrics such as BaTiO313,14,15,16,17 or in halide perovskites18,19. In the overdamped limit, the relaxation time increases with increasing damping Γ, which calls into question the picture of a well-defined phonon mode with a frequency and relaxation time. Overdamped phonon dynamics is, however, usually limited to a rather narrow temperature window, and under these circumstances, the inversion of the relationship between relaxation time and damping cannot be readily observed. Here, we demonstrate that the soft phonons modes associated with the phase transitions in the prototypical halide perovskite CsPbBr3 are, however, outstanding manifestations of this exact behavior as the overdamped region extends almost 200 K above the tetragonal-cubic phase transition.

Halide perovskites are promising materials for photovoltaic and optoelectronic applications. Specifically, CsPbBr3 has received a lot of attention in recent years20. With increasing temperature, it undergoes phase transitions from an orthorhombic (Pnma) to a tetragonal (P4/mbm) and eventually a cubic phase (Pm\(\bar{3}{{{{{{{\rm{m}}}}}}}}\))21,22,23,24,25. These phase transitions are connected to specific phonon modes and arise due to the tilting of the PbBr6 octahedra, corresponding to phonon modes at the R and M points (Fig. 1a)26,27,28,29,30. Experimentally, these modes have been shown to exhibit overdamped characteristics in the vicinity of the phase transitions18,19,31. The phase transitions have also been studied from first-principles and via molecular dynamics (MD) simulations; see, e.g., refs. 32,33,34,35.

Fig. 1: Phonons and phase transitions in CsPbBr3.
figure 1

a Phonon dispersion for the cubic phases of CsPbBr3 obtained using the machine-learned potential (MLP) in the harmonic approximation, 0 K (dashed lines) and from an effective harmonic model (EHM) at 500 K (solid lines). b Potential energy landscape along the unstable M-tilt mode calculated with MLP and density functional theory (DFT). The inset shows the CsPbBr3 crystal structure (Cs purple, Pb gray, Br red) in the energy minima, for which the PbBr6 octahedra have been tilted in-phase (visualization made with OVITO75). c–e Lattice parameters and mode coordinates obtained from a cooling run based on the isothermal–isobaric (NpT) ensemble with phase transitions at approximately 300 K and 275 K. In c–e Solid, dashed, and dotted lines refer to the Cartesian directions x, y, and z, respectively.

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Here, we reveal the dynamics of the octahedral tilt modes in CsPbBr3 over a wide temperature range via MD simulations based on a machine-learned potential (MLP) that achieves close to density functional theory (DFT) accuracy (Supplementary Note S2, Fig. S1)36,37. To obtain access to mode-specific dynamics, we project the MD trajectories onto normal modes that are associated with phase transitions in this material. As shown below, this requires both large systems (comprising at least several 10000 atoms) and sufficiently long times scales (~50–100 ns) in order to achieve converged results (see Supplementary Note S7, Fig. S9, and Fig. S8). The DFT data and the MLP models are provided as a Zenodo dataset38.

Reference data for the construction of the MLP was generated by DFT calculations39,40,41 using the strongly constrained and appropriately normed (SCAN) exchange-correlation functional42 (Supplementary Note S1). Simulations and atomic structures were handled via the ASE43 and CALORINE packages44. The phonon frequencies and relaxation times obtained with the MLP are in good agreement with experimental work for multiple phonon modes (see Fig. S5). In addition, we consider several different self-consistent phonons (SCP) methods45 as well as effective harmonic models (EHMs)46,47,48,49 using the HIPHIVE50, ALAMODE51, and SSCHA packages52.

Results and discussion

Tilt modes and phase transitions

The phase transitions in CsPbBr3, and similarly in many other perovskites, are driven by modes that correspond to tilting of the PbBr6 octahedra. These modes are located at the M (in-phase tilting) and R-points (out-of-phase tilting) in the phonon dispersion for the cubic structure (Fig. 1a). They are threefold degenerate, corresponding to tilting around the three Cartesian directions. These tilt modes exhibit a double-well potential energy surface (PES), which the MLP reproduces perfectly compared to DFT (Fig. 1b).

The MLP predicts temperatures of 300 K and 275 K for the cubic-tetragonal, Tc↔t, and tetragonal-orthorhombic, Tt↔o, transitions, respectively (Fig. 1c). This is lower than the experimental values of 400 K and 360 K20,22,23,25, a discrepancy that can be primarily attributed to the underlying exchange-correlation functional53.

The mode coordinates of the tilt modes are useful order parameters for analyzing the phase transitions (Fig. 1d, e). At 300 K, the system transitions from the cubic to the tetragonal phase, as seen in both the lattice parameters and in the freezing in of one of the three M-tilt modes (Mz). For the tetragonal phase, two R-modes (Rx and Ry) start to show larger fluctuations, and at 265 K, the system transitions to the orthorhombic phase. Here, we also note the slight difference in character between these two phase transitions. For the cubic-tetragonal transition, the order parameter (QM) and lattice parameter change sharply at the transition temperature Tc↔t (closer in character to a first-order transition), whereas for the tetragonal-orthorhombic transition, the order parameter and lattice parameter change more gradually around Tt↔o (exhibiting continuous character) in agreement with experimental observations of the transition character20,21. We note here that the mode coordinate is a global order parameter for the system. In the cubic phase, even though the mode coordinate is on average zero, there still exists a strong local correlation between the neighboring octahedra. This connects to previous work on perovskites regarding the local atomic structure deviating from the cubic structure while globally still appearing cubic28,29,54,55,56,57,58.

Mode coordinate dynamics

The mode coordinates exhibit interesting dynamical behavior already in the cubic phase far above the transition to the tetragonal phase, which can be conveniently observed in the time domain (Fig. 2a, b). At 500 K, regular (phonon) oscillator behavior is observed, whereas, at 350 K (closer but still above TC), a slower dynamic component becomes evident. Finally, at 280 K and thus below the phase transition, one observes the common oscillatory motion superimposed on a long timescale hopping motion between the two minima, corresponding to the (degenerate) tetragonal phase (Fig. 1b). We note here that the hopping frequency depends strongly on system size, and is thus not a good thermodynamic observable on its own.

Fig. 2: Phonon mode coordinates.
figure 2

Mode coordinate Q(t) for the M-tilt mode, a at 500 K (well above Tc↔t), b at 350 K (close to Tc↔t), and c at 280 K (below above Tc↔t). The M-tilt mode is three-fold degenerate (x, y, z), but here only the Mz mode is shown. Also note that for 280 K, the system switches the tilt axis at irregular intervals.

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The mode coordinate can be analyzed by fitting the respective autocorrelation functions (ACFs) to a damped harmonic oscillator (DHO) model (Fig. 3). The ACF for a regular (underdamped) mode shows a clear oscillatory pattern, as illustrated here by the highest optical mode at the R-point with a typical relaxation time of about 0.37 ps, which is longer than the mode period of about 0.2 ps (Fig. 3a). The M-tilt mode at 500 K has similar damping but is much softer (yet still underdamped), and the ACF decays with a relaxation time of about 0.58 ps (Fig. 3b). At 350 K (Fig. 3c), however, the same mode is overdamped, and in this case, the DHO model becomes the sum of two exponential decays, see Eq. (3), with relaxation times τL = 5.22 ps and τS = 0.31 ps. It is interesting to note that the decay time of the ACF at 350 K is about ten times longer than at 500 K. The DHO fits still match the data very well for both the underdamped and overdamped cases (see Fig. S11 for how the two exponential decays behave for Q and P in the overdamped case).

Fig. 3: Phonon mode auto-correlation functions (ACFs).
figure 3

ACFs (solid lines) of a the highest optical mode at the R-point as well as the M-tilt mode at b 500 K and c 350 K, along with fits to a damped harmonic oscillator (DHO) model (dashed lines). Note the large difference in time scale between b and c.

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When Γ/ω0 increases, and the system becomes overdamped, the dynamics of the modes are moving towards the diffusive Brownian motion regime. For overdamped modes, the relaxation time of the ACF increases as Γ/ω0 increases, opposite to the underdamped behavior. While this is a well-known feature of a simple one-dimensional DHO, here, one observes this behavior for phonon modes in a complex atomistic system. This phenomenon arises due to the free energy landscape being very flat close to the transition, resembling a bathtub. As a result of the high friction and weak restoring force, it, therefore, takes a long time for the DHO to move back and forth around zero (Fig. 2c; see Fig. S12 for the power spectra)59.

Frequency and relaxation times vs. temperature

The frequencies and relaxation times of the M-tilt and R-tilt modes are summarized as a function of temperature in Fig. 4. The frequency ω0 softens significantly with decreasing temperature for both modes, whereas the relaxation time τ is more or less constant. The softening of the frequency thus drives the modes to the overdamped limit with decreasing temperature. The M-tilt and R-tilt modes only become underdamped above 480 K and 410 K, respectively, well above the transition temperature to the tetragonal phase at 300 K. This indicates that we expect the phonon quasi-particle for these modes to work better at high temperatures, which interestingly is the opposite behavior compared to most phonon modes which become more damped and anharmonic with increasing temperature. At the cross-over from the underdamped to the overdamped regime, the two-time scales τS and τL emerge. When approaching Tc↔t, we see that τL increases exponentially, whereas τS → τ/2.

Fig. 4: Phonon frequencies and relaxation times.
figure 4

Frequencies and relaxation times were obtained from auto-correlation functions of a M-tilt and b R-tilt modes as a function of temperature. The shaded region indicates the overdamped regime. Note that the M-tilt mode is overdamped already about 200 K above the phase transition temperature. Here, markers are data points, and lines are interpolations to guide the eye.

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SCPs and effective harmonic models

Next, we analyze the representation of these strongly anharmonic modes by commonly used phonon renormalization techniques, specifically different SCP schemes and EHMs (Fig. 5) (see Supplementary Note S5 and Supplementary Note S6 for a more detailed description of the methods). To this end, we constructed 4th-order force constant potentials (FCPs) at each temperature which were used as input to all SCP methods; see Supplementary Note S4 and Fig. S6 for more details. There are several SCPs variants45. In SCP-alamode, the Green’s function approach is employed as implemented in the ALAMODE package51. In the stochastic self-consistent harmonic approximation (SSCHA) scheme, the harmonic free energy is minimized using gradient methods, as implemented in the SSCHA package52. In SCP-hiphive second-order force-constants are obtained by iterative fitting to forces from displacements sampled from the harmonic model and forces obtained from the MLP as implemented in the HIPHIVE package50. Here, we employ the “bare” SCP implementations in ALAMODE and SSCHA. We note, however, that there are computationally more demanding corrections for both methods35,60, the analysis of which is, however, beyond the scope of the present work. The EHMs (in this field also referred to as temperature-dependent potentials) are constructed from fitting second-order force constants to displacement and force data obtained from MD simulations with the MLP (see Supplementary Note S6 for details).

Fig. 5: Phonon frequencies comparison with SCP.
figure 5

Frequencies of a M-tilt and b R-tilt modes as a function of temperature from several self-consistent phonons (SCP) schemes, effective harmonic models (EHMs) based on molecular dynamics (MD), and damped harmonic oscillator (DHO) frequencies fitted to auto-correlation functions.

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Here, we find very similar behavior for both M-tilt and R-tilt modes. The three SCP methods (SCP-hiphive, SSCHA, SCP-alamode) employed here are in excellent agreement with each other, given the differences in theory and implementation between them. The SCP frequencies systematically overestimate the frequency ω0 obtained from the ACFs by about 1 meV (see Supplementary Note S5 for a more detailed description of the SCP methods). The EHMs constructed by fitting the forces from MD trajectories, on the other hand, show good agreement with the mode projection results. We note here that the trend for SCPs and EHMs to over and underestimate frequencies, respectively, appears to hold for all modes in the system, which is in line with previous studies35,61,62,63. However, while EHMs from MD yield a better frequency for the tilt modes compared to SCP, this is not in general true (see Fig. S7 for details).

Behavior near phase transitions

Next, we look at how these modes behave as the system goes through the transition from the cubic to the tetragonal phase. While in the cubic phase, the three M (and R) modes (denoted with subscripts x, y, and z to indicate the Cartesian direction) are degenerate, the degeneracy is broken in the tetragonal phase, and the z-direction becomes symmetrically distinct from the other two (Fig. 1c, d). Therefore, in order to distinguish these modes, we will denote them by Mxy and Mz (analogously for R-modes) in the tetragonal phase. In the tetragonal phase, there exist multiple global minima for the M-mode coordinate, as can be seen in Fig. 2c, where the system jumps between these minima. To avoid capturing this (system-size dependent) hopping time in the ACFs, we employ very large system sizes of up to 400,000 atoms, for which the system remains in the same tetragonal orientation throughout the entire simulation. Furthermore, we extrapolate the frequencies to the infinite system-size limit (Fig. S10).

The resulting frequencies are shown in Fig. 6. For the cubic-to-tetragonal transition, the M-frequency does not go to zero at the transition temperature, which is in agreement with the character of the transition being first order, as observed experimentally20,21,23. For the tetragonal-to-orthorhombic phase transition, on the other hand, the Rxy frequency does go to zero at the transition temperature, in agreement with a continuous transition as observed experimentally21. This leads to the long timescale in the DHO trending to infinity, τL → ∞, as the temperature approaches TC. Additionally, the Rxy mode exhibits a strong size dependence close to the transition temperature (see Fig. S10).

Fig. 6: Phonon frequencies across the phase transition.
figure 6

M and R-tilt mode frequencies from auto-correlation functions and damped harmonic oscillator fits as a function of temperature in the tetragonal and cubic phases. Here, filled symbols correspond to the frequencies in the cubic phase, and open symbols to the frequencies in the tetragonal phase. Solid lines show Curie-Weiss fits of the form, \({\omega }_{0}(T)\propto {(T-{T}_{{{{{{{{\rm{C}}}}}}}}})}^{p}\). For the M-mode in the cubic phase we obtain TC = 295 K (with fixed p = 0.5), while for the Rxy mode in the tetragonal phase we obtain TC = 273 K and p = 0.55. The cubic and tetragonal frequencies overlap in temperature, which is possible due to the first-order character of the transition and the fact that simulations are carried out in the microcanonical (NVE) ensemble, which does not allow for the cell shape to change between cubic and tetragonal.

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The Curie–Weiss law, \({\omega }_{0}(T)\propto {(T-{T}_{{{{{{{{\rm{C}}}}}}}}})}^{p}\), provides very good fits for the temperature dependence of the modes driving the phase transitions. For the tetragonal-to-orthorhombic transition, the fitted critical temperature, TC = 273 K, agrees very well with the observed transition temperature of 274(1) K, which is consistent with this transition being a continuous transition21. Furthermore, the fitted critical exponent of 0.55 is very close to the value of 1/2 suggested by Landau theory observed in many continuous phase transitions driven by soft modes15,64,65,66. The cubic-to-tetragonal transition has a first-order character, as evident from the finite frequency of the M mode at the transition temperature. As a result, fitting both the critical temperature and the critical exponent is ambiguous (due to the absence of data at temperatures for which the frequency goes to zero). We, therefore, fix the critical exponent to 1/2, which yields a critical temperature of 295 K, about 7 K lower than the transition temperature. Here, the critical temperature corresponds to the temperature at which the cubic phase becomes dynamically unstable, i.e., the point at which the free energy barrier between the two phases disappears.

The parameter τ remains fairly constant in the tetragonal phase across its entire temperature range for all four modes (Fig. S13). Interestingly, once the Mz mode freezes in (and the tetragonal phase is formed), both the Mz and Rz modes stiffen significantly with temperature. This results in the Rz mode becoming underdamped again with decreasing temperature at around 290 K and both M-modes approaching the underdamped limit as the system approaches the orthorhombic transition.

Conclusions

We have carried out a detailed computational analysis of the dynamics in CsPbBr3, focusing in particular on the tilt modes. We observe overdamped modes for the cubic phase almost 200 K above the cubic-to-tetragonal transition temperature. These overdamped tilt-modes exhibit correlation on very long time scales (τL) compared to the typical relaxation time (τ) or period (1/ω0) of the mode. This is in line with the dynamics of the modes transitioning toward Brownian motion due to the frequency approaching zero. What we find here is that these modes can, however, still be mathematically well described as DHOs, which allows one to formally obtain a phonon frequency and relaxation time compliant with a quasi-particle picture. A physically more intuitive description is, however, obtained if the DHO model is described by two relaxation times, which can be approximately associated with mode coordinate and momentum, respectively. As a result of the soft character of these modes, the respective amplitudes can be large already at moderate temperatures. This implies that even for relatively modest electron–phonon coupling strengths, these modes should have a notable impact on the optoelectronic properties of these materials56,67,68,69,70. A systematic investigation of these effects on a per-mode basis would be an interesting topic of further study.

In addition, we demonstrated that commonly used computational phonon renormalization methods agree very well with each other but, without extensive correction schemes, exhibit systematic errors in describing the frequencies of the anharmonic tilt modes considered here. Understanding the single-point frequencies obtained from such methods and their relation to the full dynamical spectra is thus very important when, e.g., comparing to experimental measurements.

Methods

To analyze phonon modes directly from MD simulations, we employed phonon mode projection3,71,72. The MD simulations were carried out using the GPUMD package37,73. For more details on the MD simulations, see Supplementary Note S3. The atomic displacements u(t) and velocities v(t) can be projected on a mode λ, with the supercell eigenvector eλ via

$$\begin{array}{r}{Q}_{\lambda }(t)={{{{{{{\boldsymbol{u}}}}}}}}(t)\cdot {{{{{{{{\boldsymbol{e}}}}}}}}}_{\lambda }\quad {{{{{{{\rm{and}}}}}}}}\quad {P}_{\lambda }(t)={{{{{{{\boldsymbol{v}}}}}}}}(t)\cdot {{{{{{{{\boldsymbol{e}}}}}}}}}_{\lambda }.\end{array}$$

Here, the phonon supercell eigenvector of the tilt modes were obtained with PHONOPY74, and symmetrized such that each of the three degenerate modes corresponds to tilting around the x, y, and z direction respectively. The ACFs of Q and P were calculated in order to analyze the dynamics of the modes of interest as

$${C}_{Q}(t) = \langle {Q}_{\lambda }({t}^{{\prime} }){Q}_{\lambda }(t+{t}^{{\prime} }) \rangle,$$
(1)

which can be modeled as the ACF of a DHO. The DHO is driven by a stochastic force and has a natural frequency ω0 and a damping Γ. The ACF of the DHO splits into an underdamped regime (ω0 > Γ/2) and an overdamped regime (ω0 < Γ/2). In the underdamped regime, the solution of the DHO is

$${C}_{Q}^{{{{{{{{\rm{DHO}}}}}}}}}(t)=A{{{{{{{{\rm{e}}}}}}}}}^{-t/\tau }\left(\cos {\omega }_{e}t+\frac{{{\Gamma }}}{2{\omega }_{e}}\sin {\omega }_{e}t\right),$$
(2)

where \({\omega }_{e}=\sqrt{{\omega }_{0}^{2}-\frac{{{{\Gamma }}}^{2}}{4}}\), the relaxation time is τ = 2/Γ, and A is the amplitude11. In the overdamped limit, the solution becomes the sum of two exponential decays as

$${C}_{Q}^{{{{{{{{\rm{DHO}}}}}}}}}(t)=\frac{A}{{\tau }_{{{{{{{{\rm{L}}}}}}}}}-{\tau }_{{{{{{{{\rm{S}}}}}}}}}}\left({\tau }_{{{{{{{{\rm{L}}}}}}}}}{{{{{{{{\rm{e}}}}}}}}}^{-t/{\tau }_{{{{{{{{\rm{L}}}}}}}}}}-{\tau }_{{{{{{{{\rm{S}}}}}}}}}{{{{{{{{\rm{e}}}}}}}}}^{-t/{\tau }_{{{{{{{{\rm{S}}}}}}}}}}\right)$$
(3)

where

$$\begin{array}{r}{\tau }_{{{{{{{{\rm{S,L}}}}}}}}}=\frac{\tau }{1\pm \sqrt{1 \,-\, {({\omega }_{0}\tau )}^{2}}}.\end{array}$$

Here, τS and τL denote the short and long timescales, respectively. If the natural frequency approaches zero (e.g., for continuous phase transitions driven by a soft mode), we thus expect τL → ∞ and τS → τ/2. In this limit, the resulting ACF, \({C}_{Q}^{{{{{{{{\rm{DHO}}}}}}}}}(t)\), would only consist of a single exponential decay, with a decay time approaching infinity, which corresponds to the behavior seen in Brownian motion.

Similar expressions are obtained for the ACF of the phonon velocity, which is \({C}_{P}^{{{{{{{{\rm{DHO}}}}}}}}}(t)=-\frac{{{{{{{{{\rm{d}}}}}}}}}^{2}}{{{{{{{{\rm{d}}}}}}}}{t}^{2}}{C}_{Q}^{{{{{{{{\rm{DHO}}}}}}}}}(t)\). For the overdamped case, it becomes

$${C}_{P}^{{{{{{{{\rm{DHO}}}}}}}}}(t)=\frac{A}{{\tau }_{{{{{{{{\rm{L}}}}}}}}}-{\tau }_{{{{{{{{\rm{S}}}}}}}}}}\left(\frac{1}{{\tau }_{{{{{{{{\rm{S}}}}}}}}}}{{{{{{{{\rm{e}}}}}}}}}^{-t/{\tau }_{{{{{{{{\rm{S}}}}}}}}}}-\frac{1}{{\tau }_{{{{{{{{\rm{L}}}}}}}}}}{{{{{{{{\rm{e}}}}}}}}}^{-t/{\tau }_{{{{{{{{\rm{L}}}}}}}}}}\right)$$

The ACFs for Q and P were fitted simultaneously to the DHO model in order to extract ω0 and Γ.