Evidence of large polarons in photoemission band mapping of the perovskite semiconductor CsPbBr$_3$

Lead-halide perovskite (LHP) semiconductors are emergent optoelectronic materials with outstanding transport properties which are not yet fully understood. We find signatures of large polaron formation in the electronic structure of the inorganic LHP CsPbBr$_3$ by means of angle-resolved photoelectron spectroscopy. The experimental valence band dispersion shows a hole effective mass $0.26\pm0.02\,\,m_e$, 50% heavier than the bare mass $m_0 =0.17 m_e$ predicted by density functional theory. Calculations of electron-phonon coupling indicate that phonon dressing of the carriers mainly occurs via distortions of the Pb-Br bond with a Fr\"ohlich coupling parameter $\alpha=1.82$. A good agreement with our experimental data is obtained within the Feynmann polaron model, validating a viable theorical method to predict the carrier effective mass of LHPs ab-initio.

Hybrid organic-inorganic and inorganic lead-halide perovskites (LHP) rival conventional semiconductors in multiple optoelectronic applications. LHP-based solar cells have established energy conversion efficiencies approaching 25% [1]; light-emitting devices [2] and lasers [3] are gaining considerable interest thanks to high luminescence quantum efficiency [4]. The carrier diffusion length is exceptionally long in LHPs, reaching up to several micrometers [5,6]. This property results from long carrier lifetimes, rather than from the carrier mobility [7]. While theory predicts small effective masses [8][9][10] (≈ 0.1 -0.2 m e , where m e is the free electron mass), the reported mobilities are orders of magnitude lower than in conventional inorganic semiconductors [7,11]. The microscopic mechanism underlying this unusual combination of transport properties is possibly the interplay between carriers and the ionic perovskite lattice [7,12]. In a polar crystal, longitudinal-optical (LO) phonon modes have a sizable long-range interaction with charge carriers, resulting in the formation of so-called Fröhlich polarons [13]. The polaron, heavier than a bare carrier, has a reduced mobility, compatible with the observed transport properties [12,14]. In particular, the screening of the Coulomb potential is modified in the case of polarons, purportedly explaining the observed carrier lifetimes [14,15].
The optical properties of different LHPs are known to critically depend on the details of the lead-halide bond angles [16], highlighting the importance of carrier-lattice coupling also in the photophysics of LHPs. The presence of polaron quasi-particles was indeed already proposed to model the results of several optical studies [14,15,17].
In this letter we report on experimental evidence of polaron formation by measuring its fingerprint in the electronic structure. We concentrate on the prototypical inorganic LHP CsPbBr 3 which has lately attracted interest for applications, due to better thermal and radiation stability compared to hybrid organic-inorganic LHPs [18][19][20][21][22]. The momentumresolved electronic structure of CsPbBr 3 is determined by angle-resolved photoelectron spectroscopy (ARPES) and compared with ab-initio density functional theory (DFT). Our ARPES data provide a direct measurement of the hole effective mass (m exp ) in CsPbBr 3 .
The experiment reveals a mass enhancement of 50% compared to theory, which we attribute to electron-phonon coupling. Ab-initio simulations of electron-phonon interaction show that Pb-Br stretching modes dominate the interaction. Furthermore, our calculations provide a Fröhlich coupling parameter α = 1.82, which indicate that carriers form large polarons and predict a mass renormalization in good agreement with experimental data. . The high temperature (T > 130 • C) lattice structure of CsPbBr 3 [ Fig. 1 Fig. 1 (c). Upon cooling below 130 • C, the system first undergoes a structural phase transition to a tetragonal phase, finally followed at 88 • C by a transition to an orthorhombic phase, which is the stable room-temperature lattice structure.
The structural phase transitions cause the PbBr 6 octahedra to reorient, reducing the crystal symmetry [23]. The orthorhombic phase is compared to the undistorted cubic phase in Fig. 1 (b), showing its larger real-space primitive cell and the octahedra's canting angle of approximately 10 • [11].
High-quality single crystals of CsPbBr 3 were grown from liquid solution using an inverse temperature crystallization method [24]. The CsPbBr 3 crystals were cleaved in-situ under ultra-high vacuum conditions. ARPES experiments were performed using extreme ultraviolet radiation from a high-harmonic laser source with a tunable photon energy between 20 and 40 eV [25,26]. All data were collected at room temperature, in the orthorhombic phase of CsPbBr 3 , as confirmed by X-ray diffraction [27]. To rationalize the experimental results, we performed ab-initio calculation using the Quantum ESPRESSO distribution [28,29]. The electronic structure was obtained at the generalized Kohn-Sham level using the hybrid functional scheme proposed by Heyd, Scuseria and Ernzerhof [30,31] (HSE) for the exchange and correlation energy functional. The electron-phonon interaction was accounted for within the Fröhlich model [32] with parameters obtained averaging the ab-initio Fröhlich vertex [33,34]. Further details concerning the experimental methods and the DFT calculations are given as supplemental informations [27].
The valence band (VB) photoemission intensity distribution is plotted as a function of energy and in-plane momentum wavevectors in Fig. 1 (d), for a photon energy of 37 eV. This is at odd with DFT calculations for the orthorhombic phase, which predicts an additional (back-folded) VBM at the Γ point [27]. To exclude matrix element effects and dispersion in the direction orthogonal to the sample surface (k ⊥ ), we performed energyand polarization-dependent ARPES measurements [27], which reveal no signature of an additional VBM at the Γ point. The observation of a larger k-space periodicity is not compatible with the scenario of a surface reconstruction. The additional potential associated with a periodic lattice distortion, such as that occurring in the orthorhombic phase, generally manifests itself with the appearance of back-folded bands and gaps opening at the novel Bragg planes. However, the spectral weight transfer to the novel bands is proportional to the strength of the perturbing potential and often hardly observable [35], e.g. for the methylammonium lead triiodide perovskite (MAPbI 3 ) [36,37], where no signatures of back-folded orthorhombic bands were observed by ARPES, despite a clear orthorhombic diffraction pattern.
The absence of a significant spectral weight transfer to the orthorhombic periodicity implies that the bands calculated for the cubic phase overlap well with the ARPES spectra.
The data are compared to theoretical results for the cubic phase on the right half of each panel of Fig. 1 (e) and (f). The finite experimental momentum resolution in k ⊥ , due to the short photoelectron mean free path, is accounted for by integrating the DFT bands over a range of 0.1Å −1 along the k ⊥ direction, corresponding an estimated escape depth of 5Å [38].
The material's band structure has been investigated as a function of the photon energy, and Fig. 2 shows the result for 33.5 eV, which is found to be close to the bulk R point [27]. The data correspond to the band dispersion along three high-symmetry directions (Γ−M , X −M and Γ−X) and are compared with the calculated bands in the bulk X-M-R plane. The upper valence band disperses for approximately 1.5 eV below the VBM, before reaching a deeper valence manifold, where bands are not individually resolved. Simulated element-projected partial density of states reveals that the highest-energy VB is mainly composed of Pb 6s and Br 4p orbitals derived from the PbBr 6 octahedra, in accord with previous calculations [39]. Although in the room-temperature orthorhombic phase the ARPES spectral weight follows qualitatively the DFT bands for the cubic phase, the band dispersion is modified by the structural distortion. In fact, a comparison between DFT calculations of the two phases reveals that the effective mass computed for the orthorhombic phase is 0.17 m e , higher than the cubic phase mass of 0.15 m e [27]. To determine the experimental hole effective mass, we turn to a quantitative analysis of the upper valence band dispersion which we compare with ab-initio calculations for the orthorhombic structure. ARPES data along the Γ − M direction are shown in Fig. 3. The VB energy distribution curves are well fitted by a Gaussian line shape whose width (which is not resolution-limited) is likely determined by thermal broadening with possible contributions from disorder and orthogonal momentum dispersion. To determine m exp , the valence band was fitted with a parabolic dispersion around the band maximum, until convergence was observed [27], the corresponding best fit is shown in Fig. 3. The obtained value m h = 0.26 ± 0.02 m e is in good agreement with optical measurements on CsPbBr 3 [40], where a reduced exciton mass of m exc = 0.126 m e was deduced, if one assumes balanced electron and hole effective masses, which appears justified by our DFT calculations. The effective mass calculated at the HSE level of theory for the orthorhombic phase (m 0 ) is compared to m exp in Fig. 4 (a). Theory substantially underestimates m exp , with an experimental mass enhancement of ≈ 50%, implying the presence of a mass renormalization mechanism. Comparison between HSE and G 0 W 0 effective masses shows minor changes (≈ 8%), indicating that the hybrid HSE functional gives a reasonable description of the band structure [27]. These findings seems to rule out electronic correlation effects as the main reason for the mass enhancement observed.
An important mechanism, not accounted for by the DFT calculations and relevant for polar materials, is the interaction between electrons and longitudinal optical phonons.
ARPES is sensitive to such many-body interactions, encoded in the single particle spectral function [41]. In particular, for polaronic systems, such interactions manifest themselves as a renormalization of the bare band dispersion and with the appearence of satellite peaks in the photoemission spectrum [42,43]. The satellites appear on the low-energy side of the main quasi-particle peak, at an energy separation corresponding to the relevant longitudinal optical (LO) phonon mode. In CsPbBr 3 optical phonons have a energies ≤ 25 meV [44,45], and replicas cannot be resolved within the experimental linewidth. In contrast, our analysis of the quasi-particle dispersion captures the effective mass renormalization, which we attribute to electron-phonon interaction.
This interpretation is supported by recent theoretical predictions for CsPbBr 3 and related compounds, e.g. MAPbI 3 , which exhibits the same lattice structure and similar phase diagram. Simulations of the electron-phonon interaction in MAPbI 3 predict a mass enhancement of ≈ 30 %, where the interaction is dominated by coupling with longitudinal optical phonon modes, the most important being the Pb-I stretching and bending modes, and the librational-translational modes of the methylammonium cation [46]. Since the latter modes are absent in the fully inorganic compound, we expect the largest contribution to arise from the Pb-Br bond. Simulations of hole addition into the CsPbBr 3 lattice were performed by Miyata et al. [12], showing that the largest structural relaxation occurs on the Pb-Br bond and on the Pb-Br-Pb bond angle, resulting in a reduction of the canting angle of the PbBr 6 octahedra towards the undistorded cubic lattice.
To validate this picture, we performed ab-initio calculations of the phonon bandstructure of orthorhombic CsPbBr 3 and of its dielectric function, reported in Fig. 4 (b). To estimate the Fröhlich interaction, we follow a method recently developed for polar semiconductors [34,46]. The Fröhlich vertex, which represents the matrix element for electron scattering by long-wavelength longitudinal optical phonons, can be written [33,34] as: where e is the electron charge, Ω is the volume of the unit cell, M k the mass of the atom k, Z * k the Born effective charge tensor, ε ∞ the high-frequency dielectric tensor, and ω qν and e kν (q) the eigenvalue and eigenvector associated with the mode ν of momentum q. To assess the relative importance of different phononic contributions in our calculations, the energy density of coupling d(g 2 )/dω [27] is plotted as a function of phonon energy in Fig. 4 (c).
The coupling is dominated by a maximum at an effective energy of ω LO = 18.2 meV, in the energy region of Pb-Br stretching modes [12]. The effective electron-phonon coupling to such modes is obtained integrating d(g 2 )/dω from 12 to 25 meV [see Figure 4 (b)], resulting Our calculation reveals that the coupling to the Pb-Br stretching modes is two orders of magnitude stronger compared to modes appearing in the energy range between 2 and 13 meV in Figure 4 (c), which can be associated with coupled stretching-bending modes of Pb-Br [12].
Following these calculations, we proceed to estimate the mass renormalization from the Fröhlich model [32], valid for a parabolic band dispersion and coupling to a single dispersionless LO phonon mode. In this limit, it can be shown that the coupling matrix elements g ν (q) reduces to the well-known Fröhlich coupling matrix elements [34]. The dimensionless Fröhlich coupling parameter, α, can be expressed in term of the ab-initio effective coupling strengthg 2 LO as: with m 0 the bare effective mass. We obtain α = 1.82, which fall into the small to intermediate coupling regime. In this regime, the Feynman polaron model provides a good approximation for the quasi-particle mass [46][47][48]: Here m pol is the renormalized polaron mass, and m 0 is the bare quasi-particle mass extracted from our DFT calculations. The resulting m pol = 0.24 m e is compared to the experimental result in Fig. 4. The result, in agreement with experiment within the experimental uncertainty, indicates that our model captures the main physics behind the hole quasi-particle dressing. The coupling of carriers to the Pb-Br bond modes, might play an important role also on the optical properties of LHPs, which critically depend on the Pb-Br-Pb bond angle [16]. Within the Feynman model, it is also possible to estimate the polaron binding energy and radius to be 33.5 meV and 58Å, respectively. Thus, the polaron resulting from an excess hole in CsPbBr 3 single-crystals is large, extending over several lattice unit cells. Interestingly, in the case of CsPbBr 3 nanocrystals, signatures of hole self-trapping were reported [49], suggesting that the electron-phonon interaction in LHPs nanostructures may be altered [50,51]. The adopted theoretical method can be readily generalized to multiple coupled LO phonon modes [46], as in the case of hybrid organic-inorganic LHPs.
Therefore, we expect it to be capable of predicting the carrier effective masses in the whole family of LHPs.
In conclusion, our work provides the first experimental reference for the momentum-

Experimental determination of effective masses
The VBM fit function is illustrated in Fig. S4 Fig. S4 (b). A parabolic curve was fitted for increasingly narrower regions k max ± ∆k/2 around the maximum. The corresponding effective mass, together with the fit error (plus or minus one standard deviation), are indicated in Fig. S4 (c). The parabolic model fits well the data below ∆k = 0.2Å −1 , the region was decreased symmetrically by one data point below and after the maximum, until the variation of the fit value was less than 1% for two successive steps, the chosen condition for fit convergence.
Determination of k ⊥ .
The finite photoelectron escape depth determines an uncertainty in the value of the electron momentum in the direction orthogonal to the sample surface k ⊥ . In the case of lead halide perovskites, the inelastic mean free path (IMFP) was estimated from the universal curve taking into account the presence of heavy Pb and Br atoms in reference [52]. For an IMFP of 5Å, the FWHM width the k ⊥ distribution is ≈ 0.1Å −1 , which corresponds to about ±20% of the M to R distance in reciprocal space. Under these conditions, ARPES still provides reasonable k ⊥ selectivity [53].
To determine k ⊥ we follow the evolution of the M point as a function of energy. The fitted VBM is shown in Fig. S5 (a), the corresponding theoretical dispersion is shown in   Fig. S5 (b). We used a free-electron final state model: 2m e E K cos 2 (θ) + V 0 (S1) and fitted the band dispersion with a sinusoidal function E 0 sin(2πk ⊥ ( ω)/k 0 + φ), whose periodicity k 0 was fixed to match the known lattice parameter, and those phase φ was fixed to match the theoretical energy oscillation phase. We obtain a value of V 0 = 0.7 ± 0.7 eV, from which we obtain a value k ⊥ = 0.49 ± 0.04Å −1 for the M point measured at a photon energy of 33.5 eV, close to a high symmetry plane (R point in the cubic phase).  in red: diffraction pattern from the ICSD database.

Electronic Structure calculation
All density functional theory (DFT) calculations were carried out using the Quantum ESPRESSO distribution [55,56]. We performed DFT calculation for both the cubic and the orthorhombic phase (corresponding to the room-temperature stable phase) and set the lattice parameters equal to the experimental ones [54]. The electron-ion interactions were modeled using Optimized Norm-Conserving Vanderbilt (ONCV) pseudopotentials [57] as developed by Schlipf and Gygi [58]. The electronic-structure calculations were performed at the generalized Kohn-Sham level using the hybrid functional scheme proposed by Heyd, Scuseria and Ernzerhof [59,60] (HSE) for the exchange and correlation energy functional, and including the spin-orbit coupling. An energy cut-off of 80 Ry was used for the plane-wave expansion of the wave-functions (320 Ry for the charge density) and the Brillouin zone (BZ) was sampled with a uniform Γ-centered mesh of 6 × 6 × 4 points ( 6 × 6 × 6 for the cubic phase). A reduced (density-) cutoff of 90 Ry and a grid of 3 × 3 × 2 points was used for the evaluation of the non-local component of the exchange energy and potential (the full 6 × 6 × 6 grid was used for the cubic structure).
To further check the reliability of the HSE functional, the quasi-particle band structure for the cubic phase was also evaluated within the G 0 W 0 approximation using the PBE ground state density and wave-functions as starting point and including spin-orbit coupling as implemented in the Yambo code [61,62]. Pseudopotentials including all semi-core electrons were used in this case. The parameters used for the calculation are: 80 Ry plane wave cut-off for the PBE ground state calculation, 15 Ry plane wave cut-off for the polarizability calculation, 500 bands, 1 Ry plasmon-pole energy and a 6 × 6 × 6 Γ-centered grid for the BZ integration.
Maximally localized Wannier functions [63] were computed with the Wannier90 code [64,65] and used to interpolate the HSE (and the G 0 W 0 ) band structure on an arbitrary kpoint mesh. Interpolated band structure has been used to evaluate the effective masses, as described in the next section.
The contributions of electronic states of each individual chemical element of CsPbBr 3 : Br, Cs, and Pb, to the total density of electronic states (DOS) were calculated and shown in Fig. S8.

Determination of effective mass from ab-initio DFT bands
The theoretical effective masses at the top of the valence band (R point for the cubic phase, Γ point in the orthorhombic phase) were calculated by evaluating numerically the second derivative of the Wannier-interpolated band structure ε(k).  are quite similar and differ by ∼ 8%, indicating that the HSE functional gives a reasonable description of the band structure close to the VBM.
Despite the good quality of the Wannier interpolation (see a comparison between the fully ab-initio PBE band structure and the interpolated one in Fig. S9), we point out that a small error in the absolute value of the effective masses evaluated from the interpolated bands might still be present. For the PBE functional a direct evaluation of the eigenvalues at any k point is also possible (this is not the case for the HSE functional). A comparison between the PBE effective masses obtained without the interpolation and after the interpolation is reported in Tab. I, and reveals a small overestimation of the effective masses (∼ 7%).
Overall, we are confident that our estimation of the effective masses from the interpolated band structure is correct within 0.01 m e .

Electron-phonon interaction
The electron-LO phonon (longitudinal optical phonons) interaction was accounted for within a multi-phonon Fröhlich model [66,67], i.e. assuming parabolic electronic bands and dispersionless LO phonons, and neglecting acoustic and TO phonons. The scattering by LO phonons is believed to be the most relevant process for this class of materials. [67,68] We obtained the parameters of the model averaging the ab-initio Fröhlich vertex over N q = 1000 q vectors of length 0.001 Bohr −1 uniformly distributed around the BZ center. The Fröhlich vertex is defined [69,70] as where Ω is the volume of the unit cell, M k the mass of the atom k, Z * k the Born effective charge tensor, ε ∞ the high-frequency dielectric tensor, and ω qν and e kν (q) the eigenvalue and eigenvector associated to the mode qν. All the ingredients above were computed using density functional perturbation theory as implemented in the PHONON code of Quantum ESPRESSO and using the PBE [71] functional to account for exchange-correlation effects.
The dynamical matrix has been computed in reciprocal space on a coarse grid of 4 × 4 × 4 q-point and then interpolated with standard techniques [72] and with a separate treatment of the long-range dipole-dipole interaction [73].
In Fig. (4b) of the main text the density of polar coupling [67] defined as is shown, together with the frequency dependent dielectric function in the infrared region.
The plots highlight that there is one dominant contribution at an average energy of ω = 18.2 meV. The corresponding interaction strength, averaged over the N q q-points is |g| 2 = 3.34 × 10 −5 (eV/Å) 2 . Following Ref. [66], a dimensionless parameter α ν can be defined for each relevant mode (only one in this case): with m * the hole effective mass. Inserting the effective phonon frequency and interaction strength for the unique relevant LO phonon found from the analysis above, ad using the HSE effective mass (m * = 0.171), we obtain α = 1.82, which fall into the moderate-coupling regime.