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Theory and Experiments of Pressure-Tunable Broadband Light Emission from Self-Trapped Excitons in Metal Halide Crystals

Hydrostatic pressure has been commonly applied to tune broadband light emissions from self-trapped excitons (STE) in perovskites for producing white light and study of basic electron-phonon interactions. However, a general theory is still lacking to understand pressure-driven evolution of STE emissions. In this work we first identify a theoretical model that predicts the effect of hydrostatic pressure on STE emission spectrum, we then report the observation of extremely broadband photoluminescence emission and its wide pressure spectral tuning in 2D indirect bandgap CsPb2Br5 crystals. An excellent agreement is found between the theory and experiment on the peculiar experimental observation of STE emission with a nearly constant spectral bandwidth but linearly increasing energy with pressure below 2 GPa. Further analysis by the theory and experiment under higher pressure reveals that two types of STE are involved and respond differently to external pressure. We subsequently survey published STE emissions and discovered that most of them show a spectral blue-shift under pressure, as predicted by the theory. The identification of an appropriate theoretical model and its application to STE emission through the coordinate configuration diagram paves the way for engineering the STE emission and basic understanding of electron-phonon interaction.


Introduction
Broadband light emission from self-trapped exciton (STE) in metal halide perovskites has attracted enormous attention because it can not only be used for applications such as single-component white light sources, but also provides a unique venue for studying strong electron-phonon interactions(1-5). Several reports demonstrate significant spectral tuning and intensity changes of STE emission under pressure (6)(7)(8)(9)(10)(11). However, a critical and general understanding of the effect of hydrostatic pressure on STE emission is missing. Some simply attributed it to pressure-induced effect without providing concrete understanding, while others invoked coordination configuration diagrams, but used diagrams arbitrary for cartoon illustration of their observations. There are a few first-principles calculations (FPC) of STEs in metal halide perovskites (12)(13)(14). These calculations, however, are difficult to be extended to STE under pressure in part because FPC usually fail to reproduce a non-monotonous bond length change with pressure as that observed in CsPb2Br5 (15).
Although providing insightful results, the FPC lacks the functionality of some of the parametric models that convey more transparently the pressure action on STEs. So far, there is no theoretical model that has been used to predict whether the STE spectrum in perovskites will shift to higher or lower energy under pressure. The lack of functional models prevents us from gaining insights into electron-phonon interactions and hinder potential applications of STE emission.
In this work, we first identify a theoretical model based on the standard coordinate configuration diagram that correctly describes the effect of pressure on the STE absorption line. We then present our original observations of extremely broad STE light emission in the indirect wide-bandgap CsPb2Br5 crystals and the ultra-wide spectral tuning of this emission under moderate pressure.
After identifying the free exciton (FE) emission of CsPb2Br5 in the low temperature photoluminescence (PL) spectra and having confirmed it by first-principles calculations, we establish a proper configuration coordinate diagram that provides a notable agreement between the theory and our experiment: a linear increase in STE energy with pressure and a nearly pressureindependent STE emission linewidth. Based on the model and experimental results at higher pressure, we show that at least two groups of STEs coupled to multiple phonons are involved in the STE PL emission. At the end, we review recent publications on pressure evolution of STE emission and find that nearly all of them exhibit a spectral blue-shift, in agreement with the model presented in this paper.

Modeling of STE under pressure
The theory of STE has been well elaborated (16,17), but the effect of pressure on STE emission has remained unsettled. Fig. 1a shows the standard configurational coordinate diagram that plots the adiabatic potential energy surfaces (APES) of free exciton (FE) and STE along the configurational coordinate of atomic displacements. We emphasize the importance of approximations on which an APES model is build -something that is largely missing in recent HR factors are a measure for the strength of electron-phonon coupling with phonons and in the ground and excited states. Generally, the phonon mode frequencies in the ground and excited states are not equal but in some cases they can be very close. The HR factor in the ground state also determines the PL linewidth of STE through Γ = 2.36 √ ℏ at low temperatures (18).
The energy of STE PL band center in Fig. 1(a), , is given by = − − . The energy barrier separates the FE from STE states, and is the energy needed for a transition from radiative to multi-phonon non-radiative recombination of STE. The evolution of a configurational diagram, as that in Fig. 1a, under hydrostatic pressure and the corresponding optical absorption were studied in detail in Refs. (19)(20)(21) under simplified assumptions of no change of vibrational frequencies with pressure. Here, we extend these studies to emission processes and accounting for the pressure dependence of vibrational frequencies. For details see the Supplemental Information (SI). Since the potential vibrational energy in the ground and excited states is that of harmonic oscillators, the problem of configurational diagram modification under hydrostatic pressure is essentially that of the forced harmonic oscillators. It is well known that applying a constant force to a harmonic oscillator leads only to displacement of the equilibrium position (minimum) of its potential energy well, but for a system of oscillators their relative potential energies change as well. Under the approximation in which displaces linearly with pressure p, the change in the STE PL band position ( ) at pressure p with respect to ambient, = 0, (0) is given by (20): where ( ) = ( ) 2 ( ) are the force constants of participating phonons in the ground and excited states at pressure , − masses of vibrating atoms, and ∆(0) is the -distance between the minima of and potential wells at = 0, denoted as Δ in Fig. 1(a).
(0) is the relaxation energy at zero external pressure. Under pressure , Δ changes as and the STE PL linewidth also depends on p: The second term in Eq. 1 represents one of the effects of increasing ( ) with pressure, namely the curvature of the ground state potential energy well versus increases and this alone results in decreasing ( ). We especially note this effect because it is easily predictable but can produce only red shifted ( ) with increasing pressure. The blue-shifted STE PL comes from the last two terms in Eq. 1 when they prevail over the second one.
In the simplest case, ≅ ( ≅ ) and a weak dependence of the phonon frequency on pressure ( ) ≅ (0) , shown in Fig. 1b, the STE PL band energy increases linearly with pressure due to decreasing of the self-trapped energy , while both Δ and the PL linewidth remain constant as follows from Eqs.
(1) - (3). Although these results are somewhat surprising, they comply with the so-called rigid shift theorem stating that an optical absorption line under pressure shifts in energy without change in shape if = (22). Fig. 1c presents the case, in which at pressure > 0 , the trapping phonon frequencies in the ground and excited states start to deviate and < .

Experimental Results
Large size CsPb2Br5 microplates were synthesized simply by dropping CsPbBr3 micro-powder in large quantities of deionized water in a flask (23, 24) ( Fig. 2a Fig. 2f. We assign the 423 nm PL peak to free exciton (FE) emission and the broadband to STE emission. Given that FE PL emission in CsPb2Br5 is observed only below 130K, we conclude that the corresponding 10 meV activation energy is in Fig. 1a -the barrier energy for thermally excited transitions from free to self-trapped excitons. Our previous study shows that hydrostatic pressure is very effective to distinguish STE emission from defect emission and tune the emission peak and intensity (23,(25)(26)(27), thus we applied the same diamond anvil cell (DAC) to CsPb2Br5. Fig. 3a shows a picture of a CsPb2Br5 microplate loaded in the cell. Fig. 3b shows that from 0 to 1.63 GPa, the PL intensity increases significantly.
At the same time, the PL peak experiences a large blueshift. After that, the intensity starts to decrease, while the broad light emission band continues to shift to shorter wavelength, and eventually disappears at above 5 GPa. The large PL bandwidth accompanied with the blue shifted band center makes it possible for the broad light emission to cover a wide spectral range from nearinfrared to blue. This also can be seen in the direct optical images in Fig. 3c. In particular, a warm bright white light is produced near 1.63 GPa, which is surprising and significant for an indirect bandgap crystal.   To have a better understanding of the pressure dependent PL, we also performed time-resolved photoluminescence, Raman and UV-vis spectroscopies under various pressures. Fig. 3d shows a representative PL decay curve of the CsPb2Br5 microplate at ambient temperature and Fig. 3e summarizes the change of decay time with pressure. In the same figure we compare the pressure-   (23,24) and are included in the Supplementary Online Materials. Due to the lattice compression, the Raman spectrum experiences an overall blueshift, and notably the A1g phonon at 81 cm -1 begins to separate from the nearly overlapping B1g phonon for pressures above 2 GPa (23). There is no change in the optical absorption edge under pressure below 2 GPa, and only a small redshift of ~20 meV from 2 GPa to 5 GPa is observed (Fig. S4 Supplementary Online Materials). These findings indicate that the applied pressure induces only elastic deformations of the lattice and no phase transitions and significant changes in the electronic and phonon properties are observed.

Discussions
Before comparing the theory with our experiments, we want to further prove that the 423 nm low temperature emission is due to FE radiative recombination as this is an essential parameter in the STE theory in Fig. 1. We performed the density functional theory (DFT) calculations with Heyd-Scuseria-Ernzerhof (HSE) functional and allowing for the spin-orbit coupling (SOC) (28).  To obtain a better view of the PL line shape evolution with pressure, we present in Fig. 4e the normalized experimental spectra. We note that the PL band cannot be fitted well enough with a single Gaussian lineshape as predicted by one-phonon, single-STE models (30). A splitting of the PL band into two sub-bands, however, becomes notable for pressures higher than 3 GPa and the corresponding PL band fitting results are presented in Fig. 4f. Given the splitting of the PL at pressure > 3 GPa, we believe that at lower pressure the two PL bands nearly overlap and stem from at least two different STEs coupled to phonons with similar pressure dependence of their frequencies. Apparently, the pressure dependence of STE PL is very complex and at this stage of study it would be very instructive if we can have a qualitative understanding of the experimental observation.
The pressure range of 0-2 GPa is particularly interesting because the energy bandgap changes are negligibly small and the same can be expected for the free exciton energy (23). As seen in These are the A1g phonon at 133 cm -1 involving Br -Pb stretching of Lc bond length in Fig. 4c and the B2g symmetry Pb vibrations at 69 cm -1 that corresponds to out-of-phase Br-Pb-Br bending. We believe that these or/and other non-Raman active phonons that have similar pressure dependence are creating the phonon self-trapping field of the two STEs.
In the transitional pressure range 2 GPa to 3 GPa, the PL bandwidth starts to increase while the PL band is still blue shifted. We relate this behavior to the inhomogeneous broadening as the PL energies of the two STEs start to change at different pace with pressure. Notably, a collapse of the c-axis of CsPb2Br5 crystals under pressure was observed around 2 GPa (15). This reversible isostructural phase transition is expected to have greater impact on the A1g mode at 133 cm -1 than that of the Pb B2g mode at 69 cm -1 . Indeed, with further increasing the pressure, the A1g mode frequency begins to increase rapidly, whereas the pace of frequency change of B2g mode remains low.
Above the apparent split at ~3 GPa, the energy of one of the PL bands, empty circles in Fig. 4f, remain nearly constant and the other (solid circles) start to decrease slightly. We believe that in this pressure range the phonon frequencies in the excited state begin to deviate from those in the ground state. As a result, the term proportional to 2 in Eq. 1 begins to compensate the changes due to the linear in pressure term, provided < , as demonstrated in Fig. 1c. Under same conditions, according to Eqs. (2) and (3) the PL bandwidth should decrease, which is also seen in the experimental results (empty circles) in Fig. 4f (19,21). The quantity does not depend on pressure, has dimensionality of area, but cannot be determined independently.
The diagram in Fig. 1b and Eqs.
(1)-(3) can also explain enhanced PL intensity and lifetime under initial mild pressure. As discussed above, when there is no noticeable change to both electronic and lattice structures under such a mild pressure, both the radiative and conventional non-radiative recombination rate should stay the same. The nature of STE has provided extra channels to control the dynamics and emission of excitons. The population and recombination of STE depend on two potential barriers: , which will prevent non-radiative recombination through the crossing between STE and the ground state; , which will prevent backward transition from STE to free exciton. Initial pressure will move STE upwards and result in increased PL energy. It will also make higher and thus reduce non-radiative recombination, leading to increased PL lifetime and enhanced PL intensity. However, more upward displacement of STE with increasing pressure will make it easier for STE to return back to free exciton state, leading to reduced PL lifetime and intensity.
Since the model presented in this paper has explained most of the experimental results, we use it to assess the pressure effects on STE in other perovskites. We want to emphasize that the model can reproduce any direction of the spectral shift of STE PL depending on the combination of the pressure-dependent participating phonon frequencies and their difference in the ground and excited states. We have found that the significant spectral blue-shift of STE emission under pressure is the most common phenomenon (6)(7)(8)(9)(10)(11), and it has been observed in 0D hybrid halide (C5H7N2)2ZnBr4 (8), 0D lead-free perovskite Rb7Sb3Cl16(11), 1D metal halide C4N2H14PbBr4 (7), double perovskite such as Cs2AgBiCl6, Bi-doped Cs2AgInCl6, and Cs2AgInCl6 (6,9,10). Two reports show very little or no obvious blue-shift (31,32). For example, there is almost no peak shift in corrugated 1D hybrid metal halide C5N2H16Pb2Br6 from 0-3 GPa, while at the same time, the bandgap is reduced by 160 meV (31). A possible reason for that is the self-compensating effect of a pressure increase of phonon frequencies resulting in STE energy decrease that works against the linear in pressure term in Eq. 1. Surprisingly, we have found no report on the opposite effect, i.e., spectral red shift of STE with pressure although the bandgap of most metal halides decrease under pressure. We relate this observation to the dominance of always active and increasing with pressure term ∆(0) , while the other pressure dependent terms in Eq. 1 can be zeroed.

Conclusion
In summary, we have identified a STE theoretical model and adopted it to understand evolution of STE emission in 2D CsPb2Br5 crystals under hydrostatic pressure. An excellent agreement between the model and experiments is obtained regarding the effect of pressure on STE PL peak position and peak linewidth. A careful analysis of the experimental results by using the model also reveals that two STEs are involved in the ultra-wide STE emission of CsPb2Br5. The model prediction of blue-shifted with pressure STE PL peak is also confirmed by many recently reported experiments in a variety of perovskites. This is the first model based on a general theory that help us to understand the effect of pressure and correctly predicted a major spectral change. More pressure-induced changes can be understood if more information such as specific phonons and their excited state phonon frequency is obtained. Hydrostatic pressure has been widely used to tune the STE emission, a general model will certainly help us to better design and engineer the STE for high efficiency broadband emission.

Supplementary Online Materials
Theoretical modelling, DFT, Calculation of free exciton radius, Pressure evolution of Raman and absorption spectra.