Strong Electron–Phonon Coupling Induced Self‐Trapped Excitons in Double Halide Perovskites

Double halide perovskites exhibit impressive potential for the self‐trapped exciton (STEs) luminescence. However, the detailed mechanism of the physical nature during the formation process of STEs in double perovskites is still ambiguous. Herein, theoretical research on a series of double halide perovskites Cs2B1B2Cl6 (B1 = Na+, K+; B2 = Al3+, Ga3+, In3+) regarding their electronic structures, exciton characteristics, electron–phonon coupling performances, and geometrical configuration is conducted. These materials have flat valence band edges and thus possess localized heavy holes. They also show high exciton binding energies, and their short exciton Bohr radius indicates that the spatial size of their excitons is comparable to the dimension of their single lattice. Based on the Fröhlich coupling constant and Feynman polaron radius, the stronger electron–phonon coupling strength in Ga‐series double halide perovskites is revealed. In particular, Cs2NaGaCl6 shows a high and effective Huang–Rhys factor of 36.21. The phonon characteristics and vibration modes of Cs2NaGaCl6 are further analyzed, and the Jahn–Teller distortion of the metal–halogen octahedron induced by hole‐trapping after excitation is responsible for the existence of STEs. This study strengthens the physical understanding of STEs and provides effective guidance for the design of advanced solid‐state phosphors.

the color change and self-absorption effect, which is induced by unequal lifetimes and overlapping absorption ranges of different emitters, respectively.
The formation mechanism of STEs in halide perovskites has been discussed in many studies. [22][23][24][25][26][27][28] In general, four factors are considered necessary for the formation of STEs in halide perovskites: intense exciton effect, unique atomic composition as well as the lattice structure, and strong electron-phonon coupling effect. For the exciton effect, Li et al. proposed that the STEs prefer to form in low-dimensional perovskites rather than in bulk materials. [25] This is because, in low-dimensional perovskites, the exciton binding energy inside these materials will be significantly enhanced in the direction perpendicular to the crystal layer because of the strong quantum confinement. This effect improves the existence probability of excitons in lowdimensional perovskites and then increases the corresponding self-trapping density. Other essential issues are the atomic composition and lattice structure of perovskites, which also exhibit the ability to modulate the formation of STEs. For atomic composition, Gautier et al. reported that the self-trapped emission is strong in lead and chloride perovskites but becomes weak in the iodide analog. [26] This indicates that the atomic composition influences the charge distribution as well as the geometric configuration of the lattice, finally affecting the formation of STEs. Moreover, the different arrangements of lattice arrays also have considerable influence on the self-trapping process as reported by Smith et al. [27] The relative intensity of the broadband emission in Pb-Br hybrid perovskites was found to correlate with increasing out-of-plane distortion of the Pb-Br-Pb angle in the inorganic sheets. Especially, the existence of STEs is more reported in double perovskites rather than conventional single perovskites. These newly designed Pb-free metal halide perovskites exhibit outstanding lattice tunability and electronic dimensionalities and provide ample structural possibilities for the formation of STEs. These structural features further lead to another essential factor to influence the nature of self-trapped states in halide perovskites, that is, the electron-phonon coupling effect, which is strictly affected by the atomic composition and lattice structure. [28,29] In detail, electron-phonon coupling describes the process by which the collective excitation of lattice vibration interacts with the motion of a particular electron to change its original wave function. This effect can realize strong interactions between carriers and lattices, which further induce considerable transient lattice defects to trap the primary excitons. This process is further assisted by phonons and identified by investigating the original phonon characteristics of target materials. [20,28] For example, Sui et al. have successfully developed a broadband STE emission (450-600 nm) by building a CdSe superlattice with the strong coupling of excitons and zone-folded longitudinal acoustic phonons. [28] Indeed, it is possible to realize efficient broadband emission in halide perovskites through an in-depth understanding of STEs and effective material engineering. Therefore, extensive attempts have been made in recent years to further explore the selftrapping nature of halide perovskites. By utilizing the femtosecond transient absorption (FTA) spectroscopy, Wang et al. studied the photophysical properties of different phase halide perovskites and confirmed the key ultrafast trap state capture process in the change of photoluminescence. [30] They have also successfully analyzed the potential origins of the efficient broadband photoluminescence in halide double perovskites by combining theoretical calculations and FTA experiments. [31,32] In addition, Luo et al. successfully optimized the atomic composition of lead-free halide double perovskites and obtained efficient broadband emissions. [33] However, the atomic contribution of these broadband white lights has not been pointed out directly, which is not conducive to guiding the design of advanced optoelectronic materials with STEs. The exciton effect also needs more discussion to explain its role in the formation process of STEs. The lack of this information hinders further exploration of the intriguing emission features in halide perovskites. Wang et al. discussed the atomic contribution of broadband emission in halide perovskites. [34] Although this study has well explored the origin of broadband emission in halide perovskites, the detailed electron-phonon coupling mechanism of the formation process of STEs still keeps obscure. Therefore, in-depth theoretical evidence and analysis of the formation of STEs need to be further proposed to guide the perovskites engineering for stable and efficient broadband emission.
To bridge the research gap, we aim to examine the excitonic intensity as well as the electron-phonon coupling effect in double halide perovskites to evaluate the self-trapping of excitons by utilizing the first-principle calculation. We first carried out the electronic structures of our six double halide perovskite structures to analyze their basic electronic characteristics. Their energy band diagrams are similar and exhibit some common features: the conduction band minimum (CBM) is gradient, but the valence band maximum (VBM) is flat. This reflects that they have high electron mobility as well as heavy holes. Localized heavy holes will interact with the lattice vibrations and become the prerequisite to be self-trapped. We next calculated the exciton binding energy as well as the exciton Bohr radius of these double halide perovskites by applying the hydrogenic Rydberg model. In general, structures exhibiting high-exciton binding energy and low-exciton Bohr radius are considered to have a significant exciton effect and are selected for further investigation. We then use Fröhlich coupling constants to describe the electron-phonon coupling strength in these structures and calculate their corresponding polaron radii. The energy of self-trapping (E st ) and lattice deformation (E d ), which describe the energy differences of excited and ground states between self-trapping and free configurations, is determined in the configuration coordinate diagrams and is used to estimate the Huang-Rhys factors (S) of these lattices. The large Huang-Rhys factor (S ¼ 36.21) confirms the strong electron-phonon coupling in double halide perovskite Cs 2 NaGaCl 6 and the following electron density difference diagram points out the distribution of its STEs. The interaction between electrons and collective lattice vibration causes the GaCl 6 octahedron in the excited state to trap a hole, which changes the electronic configuration of Ga 3þ to 3d 9 . Split energy levels in the crystal field break the spatial symmetry to reduce the degeneracy and cause Jahn-Teller distortion of the GaCl 6 octahedron, which is considered as the origin of STEs ( Figure 1). Our study provided effective analysis as well as methodology in revealing the intrinsic physics of STEs in double halide perovskites, which contributes to guiding further engineering of double halide perovskites in solid-state photoluminescence.

Results and Discussions
In this study, we focused on the double halide perovskites Cs 2 B 1 B 2 Cl 6 (B 1 = Na þ , K þ ; B 2 = Al 3þ , Ga 3þ , In 3þ ) to explore the connections between STEs and their lattice structures. Traditional metal halide perovskites, such as CsPbCl 3 , show impressive photovoltaic performance due to their high electronic dimensionality and strong antibonding between lead and halogen orbitals. [35][36][37] However, these emerging materials also suffer serious challenges of their instability at high humidity as well as temperature and toxicity of lead. The combination of trivalent and monovalent cations on the B-sites of traditional metal halide perovskites to form double halide perovskites is an obvious choice to achieve lead-free lattice and the corresponding structural tunability. Also, as we mentioned above, the enhanced structural tunability of these double halide perovskites can contribute to the formation of STEs to produce broadband emissions. Here, we have chosen the double halide perovskite materials consisting of monovalent (B 1 = Na þ , K þ ) and trivalent cations (B 2 = Al 3þ , Ga 3þ , In 3þ ) belonging to the same main group for a brief comparison. Experimentally, some of these materials have been synthesized and exhibited great potential in STEs photoluminescence. [38][39][40] In this way, we aim to realize the strong STE effect in these materials and achieve an effective exploration of their STE properties with regular atomic compositions. As shown in Figure 2a  www.advancedsciencenews.com www.advenergysustres.com (B 1 = Na þ , K þ ) and trivalent cations (B 2 = Al 3þ , Ga 3þ , In 3þ ) occupy the original B-sites alternately, forming a corner-sharing 3D array of metal-halogen octahedrons. The optimized lattice structures of our six samples are similar and the representative Cs 2 NaGaCl 6 lattice structure is shown in Figure 2b. The lattice belongs to the triclinic system and the two different kinds of octahedrons, NaCl 6 and GaCl 6 , are aligned alternately. The basic lattice information of these materials including lattice lengths and bonding angles are presented in Table S1 and S2, Supporting Information. Our calculated bandgap values of these double halide perovskites have shown good consistency with previous works (see Table S3, Supporting Information).
The electronic structure of semiconductors contains key optoelectronic information to elucidate the origin of STEs. We first calculated the electronic structures of these six samples (see Figure S1, Supporting Information) and the energy band diagram of Cs 2 NaGaCl 6 is shown in Figure 2c. It is worth noting that its CBM exhibits a gradient band edge with broad bandwidth and the band edge of VBM is flat. The gradient conduction band edge reflects the dispersive features as well as the high electron mobility of this material, which contributes to the high electronic dimensionality. [35] In contrast, the flat VBM represents the low mobility of the holes. Through investigating the density of states, it is noted that the valence band edge is mainly contributed by the GaCl 6 octahedron. Alternating insertion of NaCl 6 octahedrons breaks the orbital continuity of the original 3D GaCl 6 octahedral connection networks, resulting in decreased hole mobility. These heavy holes are difficult to diffuse and gradually localized, finally improving the corresponding effective hole mass. The localized holes further interact with surrounding lattices frequently and are easily self-trapped, which is the prerequisite for the formation of STEs.
Strong exciton characteristics also play an important role in the formation process of STEs, which could be well described by exciton binding energy (E b ). We next tested the exciton characteristics inside these materials to investigate the STEs features. In general, photogenerated excitons can be regarded as a lone electron-hole pair, and thus its binding energy can be represented by exciton Rydberg energy and estimated by applying the hydrogenic Rydberg model [41] where R y is the Rydberg unit of energy, ε r is the relative permittivity, m 0 is the electron mass, and μ ¼ ðm Ã e m Ã h Þ= ðm Ã e þ m Ã h Þ represents the reduced mass of the electron-hole pair. Based on the energy band information of these double perovskites, we further obtained the effective masses of electrons (m Ã e ) and holes (m Ã h ) at the band edges by calculating The above calculation results of all double halide perovskite are completely presented in Table 1. Based on these results, we can identify that the effective mass of holes in these materials is about eight times higher than electrons, consistent with the low hole dimensionality reflected by the valence band edges. As for the permittivity, one issue worth elucidating is that the selection of relative permittivity for the calculation of E b is still controversial. [42] Frequency-dependent relative permittivity mainly describes the ability of dielectric material to shield the external instantaneous electric field by its internal polarization, which consists of electronic and ionic contributions. The true effective dielectric constant should be a value intermediate between the lowfrequency static (ε static ) and high-frequency optical (ε ∞ ) dielectric constants. In halide perovskites, there is always a large difference between these two dielectric constants, indicating that an arbitrary choice leads to an incomplete result. [42] Therefore, we calculated the upper and lower bounds of E b and presented the corresponding range of different objects in Figure 3a. Another parameter, exciton Bohr radii, which describe the spatial size of the corresponding exciton, can also be calculated by applying formula r b ¼ m 0 ε r μ a H . Here, a H represents the Bohr radius of the hydrogen atom. The range of exciton Bohr radii of these perovskites is also shown in Figure 3b.
As shown in Figure 3a, the lower bound E b of these perovskites, which is based on the static dielectric constant ε static , is similarly distributed in the range of 100-200 meV. Cs 2 NaAlCl 6 and Cs 2 KAlCl 6 have the highest binding energy of %185 meV as the lower bound E b in their corresponding series. Such high values indicate that these double halide perovskite materials have considerable exciton effects compared with the traditional halide perovskites, which have generally larger static dielectric constants (ε static % 30) and lower E b (%30 meV). [43][44][45] Meanwhile, the high values of the lower bound E b of these materials further ensure the basic exciton conditions for the formation of STEs. The upper bound E b of these double halide perovskites, however, exhibits differences between Na-and K-series materials. Naseries double halide perovskites keep the higher bound E b of %670 meV, while the value of is Cs 2 KGaCl 6 even more than 900 meV. Compared with Na-series samples, K-series double halide perovskites have higher reduced mass μ and lower optical dielectric constant ε ∞ as shown in Table 1, respectively. This suggests that there is a relatively stronger lattice periodic potential field and lower electronic polarization response in these K-series structures and thus, stronger electron-lattice scattering. Interactions between carriers and lattice enhance the quantum confinement of electron-hole pairs after excitation, which is responsible for the higher upper bound E b of these materials. This judgment also applies to Na-series double halide perovskites above when the comparison objects are traditional halide perovskites. The exciton Bohr radii r b of these double halide perovskites are also estimated based on the presented data. In general, these values reflect the degree of spatial confinement of electron-hole pairs in crystals and are positively correlated with the exciton effect. It can be seen that the exciton Bohr radius r b of these double halide perovskites is less than 10 Å in general, which is less than the longest diagonal distance r L of their triclinic lattices (see Table S2, Supporitng Information). The possible spatial dimensions of excitons in the corresponding lattice of these materials are shown in Figure 3b. Considering the lower bound r b , these materials even possess highly localized excitons of %3 Å size inside. This indicates that the excitons of these double halide perovskites are strong Frenkel excitons and are able to trigger significant exciton-lattice coupling to fulfill the prerequisite of the formation of STEs. After confirming the intense excitonic environment in these double halide perovskites, we further explored the electroncoupling effect. In theory, electrons or holes in ionic crystals polarize their neighbor lattices. Then, the moving carriers can create distortions in lattices and move together as a unified polarization state, which has been called polarons. By assuming that all the essential phonons have the same frequency in the dielectric crystal lattices, Fröhlich reduced the problem of moving electrons in the lattice potential field to the following Hamiltonian [46] H ¼ where r el represents the position vector of the electron and P is the conjugate momentum. w is the momentum, b þ w , b w represent the corresponding creation and annihilation operators of a phonon, respectively. V ¼ L 3 describes the cube volume and . A dimensionless variable called Fröhlich coupling constant α can hence be extracted from the above equation to describe the electron-lattice interaction strength C s 2 N a A lC l6 C s 2 N a G a C l6 C s 2 N a I n C l6 C s 2 K A lC l6 C s 2 K G a C l6 C s 2 K I n C l6 where ω LO represents the characteristic phonon angular frequency of corresponding materials and the dielectric contrast term 1 ε ∞ À 1 ε static effectively quantifies the carrier-lattice interaction strength in ionic crystals. In our double halide perovskites, there are multiple phonon branches coupled with electrons and so we calculated one effective longitudinal optical phonon frequency by taking a spectral average of all infrared active optical phonon branches [47] W 2 e Ω e cothðβ e =2Þ ¼ where Ω is the LO frequencies and W is oscillator strength. To further show the polaron characteristics of these double halide perovskites, we evaluated the corresponding radius and effective mass of polarons for small α by considering the Feynman polaron theory [48,49] r p ¼ ð3=0.44αÞ Here, we present the calculation results of the above parameters in Table 2 and compare the coupling constant α as well as the polaron radius r p of different double halide perovskites as shown in Figure 3c. These double halide perovskites exhibit considerable Fröhlich coupling constants of %3-5, which is higher than the average level of traditional halide perovskites (%2). [50] This indicates that these double halide perovskite structures composed of alternately different metal-halogen octahedron arrangements have a stronger electron-phonon coupling effect. Such a strong electron-phonon coupling effect assists the localized carriers in low electronic dimensionality to interact with the lattice frequently, creating more possibilities for the generation of STEs. In detail, the variation of the coupling constants is related to the collocation degree between B 1 and B 2 octahedra: smaller B 2 octahedron increases the coupling constant in Na-series double halide perovskites but decreases the coupling constant in K-series ones. Meanwhile, it can be seen that the coupling constant values correlate well with the upper bound E b of corresponding materials, which are mainly determined by the optical dielectric constant ε ∞ , as mentioned above. This further verifies the interaction between electrons and long-wave polar optical phonons.
The concept of self-trapping originates from the initial sense of polaron, that is, the electron interacting with long-wave polar optic vibrations. When excited electrons keep the inevitable coupling effect with their neighbor lattice vibrations, they can be considered in a self-trapped state, which also applies to excitons. [51] Therefore, polaron properties do counts for the formation process of STEs in crystals. In our study, the polaron radii of these double halide perovskites keep in the range of 60-90 Å. This spatial dimension matches well with 2-3 times the diagonal distance r L within the lattice. Cs 2 NaGaCl 6 and Cs 2 KGaCl 6 own the shortest polaron radius in their corresponding series: 74.46 and 59.80 Å, respectively. Such a short polaron radius reflects the ample influence of lattice vibrations on the mobility of carriers, which induces high effective polaron mass (as shown in Table 2). In addition, the shorter exciton Bohr radius in our materials as mentioned above implies that these polarons, or self-trapped states, are more likely to be caused by Frenkel excitons rather than individual electrons. The distribution of polaron radii of these double halide perovskites, however, is slightly different from the coupling constants after considering the effective mass of electrons m e and LO frequencies ω LO . As shown in Figure 2c, the Ga-series double halide perovskites, Cs 2 NaGaCl 6 and Cs 2 KGaCl 6 , show shorter polaron radii than their analogs, which perform more potential of the existence of STEs.
Since the generation of STEs requires a strong electronphonon coupling effect, this whole process will definitely lead to the corresponding lattice configuration and energy changes, especially in the excited states. Therefore, we then carried out the configuration coordinate diagrams of these double halide perovskites ( Figure S2, Supporting Information) by first computing the coordinate difference as ΔQ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P where k, M, and R represent the atom, atomic mass, and the position coordinates for excited (E) and ground (G) states, respectively. [33] d ¼ ðx, y, zÞ denotes different directions. Complete coordinate Q was determined by linear interpolation based on the obtained ΔQ. It can be seen that the optimized excited nuclear configuration has lower potential energy when compared to its primary state and the energy difference is attributed to the self-trapping process (E st ). Meanwhile, this configuration also increases the potential energy of the original ground state and causes corresponding lattice deformation (E d ). In general, the energy component of the whole emission can be regarded as E g ¼ E st þ E PL þ E d and the computed data can be found in Table S4, Supporting Information. The diagrams of Cs 2 NaGaCl 6 and Cs 2 KGaCl 6 are presented in Figure 4a,b, respectively. Compared to their analogs, Cs 2 NaGaCl 6 and Cs 2 KGaCl 6 have obviously higher E st and E d , which match well with their polaron radii characteristics. Smaller polaron radius provides a higher energy barrier for carrier self-trapping and increases E st , which contributes to the formation of stable self-trapped states. Meanwhile, the lattice deformation energy E d is positively correlated with the self-trapping energy E st , where a high E d value describes a considerable lattice distortion inside these double halide perovskites. This connection implies that the lattice distortion caused by electron-phonon coupling is the exact source of STEs formation. The Huang-Rhys factor S ¼ E d ℏΩ LO has been estimated to evaluate the strength of the STE effect in these double halide perovskites. [52] As presented in Table 3, all of these structures exhibit a noble electron-phonon coupling effect (see Figure S2, Supporting Information) and Cs 2 NaGaCl 6 has the highest S value of 36.21, which far exceeds the traditional halide perovskite crystals. [53,54] Meanwhile, the sufficient separation between the curves of the ground and excited state indicates that this high S value does not induce the nonradiative recombination between the electrons and holes in Cs 2 NaGaCl 6 . Therefore, Ga-series double halide perovskites, especially Cs 2 NaGaCl 6 , show potential to be promising candidates for the STEs materials and be further applied in the solid-state photoluminescence field. The electron density difference of excited Cs 2 NaGaCl 6 after optimization is also visualized in Figure 4c to explore the STEs situation within the lattice. The blue and yellow areas represent the spatial density of electron and hole orbitals, respectively. It is clear that electrons tend to extend toward the vertices of the GaCl 6 octahedrons due to the strong bonding characteristics and the unique spatial relative position between Ga and Cl atoms. In contrast, the hole orbitals shrink into the center of the GaCl 6 octahedrons. This type of geometric confinement, which is consistent well with the large differences of effective mass between electrons and holes as shown in Table 1, will hinder the migration of holes and thus increase the possibility of their self-trapping.
Phonon characteristic is another aspect that needs to be discussed for these structures. In polar semiconductors such     [53] CsPbBr 3 (microcrystals) 12.00 [54] www.advancedsciencenews.com www.advenergysustres.com as our double halide perovskite structures, atomic vibrations because of the long-wavelength LO phonons will result in macroscopic polarization and thus an internal electric field, which further enhances the Coulomb interactions between the transferring carriers and crystal lattices. [55] For Cs 2 NaGaCl 6 , there are 10 atoms in one unit lattice, leading to 30 vibration modes in total, including 3 acoustic branches, and 27 optical branches. Its point group is calculated as O h and the sum symmetry modes at the center of the Brillouin zone can be described with irreducible representations as Γ ¼ 10E u þ 5A 2u þ 9T g þ 3T u þ 2E g þ A 1g , in which 5E u þ 6T g þ 2T u þ E g are degenerate. Among these vibration modes, 8E u þ 4A 2u þ 3T u are infrared active modes and 9T g þ 2E g þ A 1g Raman active modes. Three acoustic modes, 2E u þ A 2u , are silent because of the lack of quadratic functions. [56] We then proposed the diagrams of phonon dispersion curves and phonon state density of Cs 2 NaGaCl 6 in Figure 5a. It can be seen that the LO phonon frequencies are roughly distributed in three ranges: 40-80 cm À1 , 100-20, and 220-270 cm À1 . For the low-frequency LO phonons, Cs atoms contribute the most and the wagging between Cs and Cl atoms becomes the main vibration source. For the higher frequency range 100-200 cm À1 , which is specified with a pink area as shown in Figure 5a, interactions between Cl and two kinds of metal atoms occupy the most phonon states. The high-frequency stretching between Ga and Cl atoms is responsible for the scarce phonon states in 220-270 cm À1 . As discussed above, the calculated effective LO frequency of Cs 2 NaGaCl 6 is 131.61 cm À1 . Therefore, the main LO phonon branches in this structure, which interact with the moving carriers in the process of electron-phonon coupling, should be closely related to the atomic vibrations between different metal-halogen octahedrons.
To further investigate the exact coupling results between different metal-halogen octahedrons and analyze the origin of STEs, the bond lengths of metal octahedrons in www.advancedsciencenews.com www.advenergysustres.com