The interaction between electrons and phonons in CsPbBr3 is essential for the design of CsPbBr3 based optoelectronics since the phonon governs emission line broadening in metal halide perovskites. In this work, the temperature dependent photoluminescence linewidth was obtained from 80 K to 400 K. Density functional theory and many-body perturbation theory were used to calculate the electron–phonon coupling in CsPbBr3. We demonstrate that the electron–phonon coupling is dominated by the longitudinal optical phonon. In addition, the photoluminescence spectrum broadening is almost linear with temperatures from 80 K to 400 K. Their results provide a better understanding of the mechanism of electron–phonon scattering in CsPbBr3 and related optoelectronic devices.

In the past few years, lead halide perovskites have attracted much attention in the field of solar cells based on perovskite thin films due to their high power conversion efficiency, such as MAPbI3 and MAPbBr3 (MA: CH3NH3).1–3 Recently, it has been found that lead halide perovskites exhibit excellent luminescence properties, which are suitable for light emitting diodes (LEDs). In addition to the organic–inorganic methylammonium lead halide perovskites, another special variation of the lead halide perovskite, such as CsPbBr3, is all-inorganic cesium lead halide perovskites. CsPbBr3 also exhibits superior luminescence properties, such as narrow emission spectra and high luminescence efficiency.4 Compared with MAPbI3 and MAPbBr3, CsPbBr3 is more stable in air,5 which makes CsPbBr3 a competitive candidate for LEDs6 and photodetectors.7 The success of lead halide perovskites in photovoltaic and luminescent applications has been attributed to their low exciton binding energy8 and long charge-carrier diffusion length.9 

Since the electron–phonon interaction plays an important role in the fundamental intrinsic limit of charge-carrier mobility and charge-carrier cooling following non-resonant photon absorption, many efforts have been devoted to reveal the charge-carrier recombination mechanisms underlying these properties.10–12 However, the interaction of charge carriers with phonons is still under debate. In order to address this issue, several groups have examined the temperature dependence mobility,10,11,13,14 which is only governed by deformation potential scattering with acoustic phonons.15 They found that the mobility is almost proportional to T1.5, which is close to the theoretical result of electron–phonon coupling at room temperature. So far, although a lot of work has been done on the electron–phonon coupling in organic–inorganic hybrid lead halide perovskites,16–19 there are still few studies on all-inorganic lead halide perovskites such as CsPbBr3. In this work, the experimental photoluminescence (PL) linewidth of CsPbBr3 as a function of temperature has been obtained between 80 K and 400 K in order to investigate the charge-carrier scattering mechanism in inorganic lead halide perovskites. At the same time, the theoretical PL linewidth was derived from ab initio calculation for CsPbBr3 by Quantum ESPRESSO,20 YAMBO,21 and EPW.22 We further demonstrate that the experimental results are in good agreement with the theoretical results. Both results indicate that the Fröhlich coupling to the LO phonon is the main reason for the linewidth broadening in inorganic lead halide perovskites.

The CsPbBr3 sample was prepared by the solution-based method.23 First, the CsPbBr3 sample was mounted with a pressure lower than 1 × 10−6 mbar in a liquid nitrogen cryostat under vacuum. A relevant temperature sensor was installed at the end of the sample holder to monitor the temperature. The CsPbBr3 sample was photoexcited with a 266-nm laser. The PL was collected and focused into a grating spectrometer. The sample was heated between 80 K and 400 K in 5 K increments, while PL measurement was performed at the same time.

The ground state electronic structure was calculated by the Quantum ESPREESO (QE) package within the local density approximation including spin–orbit coupling.20,24 The optimized norm-conserving Vanderbilt pseudopotential (ONCVPSP25) was used to describe the core–valence interaction, which is generated by the PseudoDojo package.26 The plain wave self-consistence field calculation converged with a plane-wave cutoff of 100 Ry and a 6 × 6 × 6 Brillouin-zone grid. The phonon calculation was performed within density functional perturbation theory with a 2 × 2 × 2 Brillouin-zone grid by the PHonon package in QE.20,24 The definition of the k-point and q-point labels was defined by the SeeK-path tool.27 The electronic Quasi-Particle (QP) energy was obtained by the GW method using the YAMBO code.21,28 After GW calculation, the band structure details of the valence band and conduction band were further interpolated by maximally localized Wannier functions using Wannier90.29 The electron–phonon coupling was further calculated by the EPW package.22 The convergence of the electron–phonon self-energy required up to one million random phonon wavevector q-points in the Brillouin zone.

The cubic and orthorhombic structures of CsPbBr3 are shown on top of Figs. 1(a) and 1(b), respectively. The ab initio calculations on CsPbBr3 in the cubic phase were carried out using the crystallographic data obtained by the neutron diffraction technique,30 and the ab initio calculations in the orthorhombic phase were carried out using the crystallographic data obtained by x-ray diffraction.31 Both calculation structures were fully relaxed, including the lattice constants, until the force was less than 2.5 × 10−4 eV/Å. The lattice constants are shown in Table I. Comparing the calculation lattice constant with the experiment lattice constant, the relative error is within 3%. Furthermore, the unit cell size of the orthorhombic structure is approximately 2×2×2 times the size of the cubic unit cell. Compared with the highly symmetric cubic phase, the displacement of atoms in the orthorhombic phase is very small. Yang et al. already showed that the cubic phase of metal halide perovskites had been used extensively that could ensure good estimation of electron and phonon properties.32,33 The phonon DOS (Density of States) of CsPbBr3 in cubic and orthorhombic phases are shown in Figs. 1(a) and 1(b), respectively. The calculation results indicate that the phonon DOS of these two states are comparable. In particular, we have identified three main peaks in the cubic and orthorhombic phase phonon DOS, which can be directly compared. These peaks are marked in Figs. 1(a) and 1(b). The phonon band structures of the cubic and orthorhombic phase of CsPbBr3 are shown in Figs. 1(c) and 1(d), respectively. The soft phonon modes with a negative frequency in Fig. 1(c) result from the metastable lattice of the perovskite cubic phase, where a permanent displacement of the atoms is possible through the distortion of the cubic crystal lattice.34 These soft phonon modes with a negative frequency will be ignored for the further electron–phonon interaction calculation. On the other hand, the unit cell of CsPbBr3 in the cubic phase is composed of only 6 atoms, while the unit cell of the orthorhombic or tetragonal phase is composed of 24 atoms. Generally, the computational workload will increase exponentially as the number of atoms in the unit cell increases. Therefore, the computational workload of CsPbBr3 in the cubic phase will be far less than the other two phases, especially for the electron–phonon interaction computation using EPW, which requires one million random phonon wavevector q-points to ensure convergence.22 Hence, all the electron–phonon coupling calculations of CsPbBr3 were only performed in the cubic phase in this work. However, more work on the electron–phonon coupling calculation in orthorhombic and tetragonal phases needs to be done in the future, in order to study the detailed difference between different phases.

FIG. 1.

The phonon calculation results of CsPbBr3. (a) and (b) are the phonon density of states of CsPbBr3 in cubic and orthorhombic phases, respectively. At the top of (a) is the top view of the cubic structure, and at the top of (b) is the top view of the orthorhombic structure. (c) and (d) are the phonon band structures of CsPbBr3 in cubic and orthorhombic phases, respectively.

FIG. 1.

The phonon calculation results of CsPbBr3. (a) and (b) are the phonon density of states of CsPbBr3 in cubic and orthorhombic phases, respectively. At the top of (a) is the top view of the cubic structure, and at the top of (b) is the top view of the orthorhombic structure. (c) and (d) are the phonon band structures of CsPbBr3 in cubic and orthorhombic phases, respectively.

Close modal
TABLE I.

Lattice constant of CsPbBr3.

CubicOrthorhombic
a (Å)a (Å)b (Å)c (Å)
Experiment30,31 5.605 8.26 11.76 8.21 
Calculation 5.78 8.24 11.73 8.20 
CubicOrthorhombic
a (Å)a (Å)b (Å)c (Å)
Experiment30,31 5.605 8.26 11.76 8.21 
Calculation 5.78 8.24 11.73 8.20 

The temperature dependent steady-state PL spectra were recorded from 80 K to 400 K in increments of 5 K, which are shown in Fig. 2. Figure 2(a) shows the pseudo color plot of PL spectra with the temperature range from 80 K to 400 K, while Fig. 2(b) shows the normal plot of PL spectra at the selected temperature. The black dotted lines in Fig. 2(b) are the fitting results of the Gaussian line shape. Comparing the experimental PL spectrum with the Gaussian fitting result, the line shape is a little asymmetric at lower photon energies. Since the PL spectrum displays conduction band to valence band transition,35 the asymmetric line shape can be due to the asymmetric density of states of CsPbBr3 around the bandgap. The bandgap at room temperature is about 2.3 eV, which is consistent with the previous results.36 Compared with the PL spectra of MAPbBr3, which exhibit an abrupt shift in PL peak energy at about 150 K,18 the PL spectrum peak energy of CsPbBr3 exhibits a steady increase as the temperature increases. The abrupt shift in the PL peak energy of MAPbBr3 is mainly associated with the phase transition of the orthorhombic phase to tetragonal phase because the extent of rotational freedom of the MA+ cation is strongly reduced during phase transition.37 Although CsPbBr3 undergoes phase transition from the orthorhombic to tetragonal phase at about 361 K and phase transition from the tetragonal to cubic phase at about 403 K,38 we could not observe any abrupt shift of the PL peak in Fig. 2(a) because there is no reduction in the rotational freedom of the Cs+ cation during the phase transition. Furthermore, the bandgap of CsPbBr3 increases with temperature, which indicates that the bandgap deformation potential is positive. Such a result is similar to the situation of MAPbBr3, which results in the stabilization of out-of-phase band-edge states as the lattice expands.39 Moreover, we could not observe any obviously banggap change at 361 K, which is the phase transition temperature as indicated by Sakata et al.30 

FIG. 2.

Temperature dependent steady-state PL spectra. (a) is the color plot of PL spectra of CsPbBr3. All spectra in (a) are normalized. The color bar indicates the intensity of the spectra. The yellow in (a) means higher intensity, and the purple in (a) means lower intensity. (b) is the normal plot of the PL spectrum at the selected temperature. The black dotted lines in (b) are the fitting results of the Gaussian peak shape. (c) is temperature dependent bandgap of CsPbBr3 extracted from the temperature dependent PL spectra.

FIG. 2.

Temperature dependent steady-state PL spectra. (a) is the color plot of PL spectra of CsPbBr3. All spectra in (a) are normalized. The color bar indicates the intensity of the spectra. The yellow in (a) means higher intensity, and the purple in (a) means lower intensity. (b) is the normal plot of the PL spectrum at the selected temperature. The black dotted lines in (b) are the fitting results of the Gaussian peak shape. (c) is temperature dependent bandgap of CsPbBr3 extracted from the temperature dependent PL spectra.

Close modal

The bandgap of CsPbBr3 can be simply approximated by a linear function as

(1)

where E0 = 2.25 eV and αg = 1.4 × 10−4 eV/K. Such a linear temperature dependent bandgap has already been observed in CsSnI3.40 The red line in Fig. 2(c) is the fitting curve, while the orange band around the red line is the confidence band of the fitting result with a 3σ confidence interval.40,43 The fitting results are calculated by the LMFIT package, which is a non-linear least-squares minimization and curve-fitting tool for Python.41 Comparing the experimental data with the fitting results, it is found that some experimental data are beyond the confidence band, especially the data in the temperature range between 200 K and 250 K. This deviation between experimental data and theory predictions may be due to the instability of experimental equipment, including spectrometers, lasers, and heating and cooling sample stages, because each measurement between two temperature points must be separated by at least 10 min until the temperature stabilizes. The entire PL spectra were measured for more than 10 h. So far, it is still a challenge for us to meet the requirements of long-term PL spectrum measurement stability or fast stable temperature stabilization time.

Both thermal expansion and electron–phonon interaction contribute to the temperature dependence of the bandgap. Sometimes, the effect of electron–phonon interaction on the bandgap can be very large.42 In order to investigate the effect of thermal expansion on the bandgap of CsPbBr3, we used DFT to calculate the bandgap of CsPbBr3 in the cubic phase, where the lattice constant is in the range of a0 ± 0.01a0. a0 is the calculated equilibrium lattice constant shown in Table I. The DFT calculation results show that the theoretical bandgap change is almost linear with the change in the lattice constant, which can be expressed as

(2)

where β = 2.36 eV/Å. The linear thermal expansion coefficient αL of CsPbBr3 in the cubic phase is about 3.3 × 10−5 K−1.43 Therefore, we could estimate the bandgap change due to the thermal expansion, which can be approximated by

(3)

where αg,DFT = αLβa0 = 4.5 × 10−4 eV/K. The DFT calculated coefficient αg,DFT is three times larger than the measured coefficient αg. The results indicate that the electron–phonon coupling can play an important role in the bandgap renormalization of CsPbBr3. The effect of electron–phonon coupling on the bandgap renormalization of the CsSnI3 has already been proved in detail by Patrick et al. using theoretical calculation.44 In order to calculate the bandgap renormalization of CsPbBr3, there are several methods, such as Allen–Heine–Cardona theory45,46 or the finite displacement method.47 However, the detailed calculation of CsPbBr3 bandgap renormalization is beyond the scope of the present work. More work on the effect of electron–phonon coupling on the bandgap renormalization in CsPbBr3 is definitely needed and will be done in the near future.

The temperature dependent linewidth of semiconductor PL spectra has been used to study the interaction between electrons and phonons.48 In Fig. 3, a heat-map of the imaginary part of the electron–phonon self-energy [Im(σ)] at 300 K projected on the quasiparticle band structure of CsPbBr3 is presented. The imaginary part of the electron–phonon self-energy corresponds to the linewidth of electrons and holes due to electron–phonon interactions. Therefore, the imaginary part of the electron–phonon self-energy can be directly used to predict the PL linewidth.

FIG. 3.

Electron–phonon coupling calculation results of CsPbBr3 by EPW.22 The electronic band structure of cubic CsPbBr3 is calculated within the GW approximation21 and is interpolated by the Wannier function using Wannier90.29 The heat map indicates the imaginary part of the electron–phonon self-energy [Im(σ)] at T = 300 K. The zero energy is located in the middle of the bandgap at the high symmetry point R. The calculated bandgap from GW approximation is about 2.25 eV.

FIG. 3.

Electron–phonon coupling calculation results of CsPbBr3 by EPW.22 The electronic band structure of cubic CsPbBr3 is calculated within the GW approximation21 and is interpolated by the Wannier function using Wannier90.29 The heat map indicates the imaginary part of the electron–phonon self-energy [Im(σ)] at T = 300 K. The zero energy is located in the middle of the bandgap at the high symmetry point R. The calculated bandgap from GW approximation is about 2.25 eV.

Close modal

In order to extract the temperature dependent PL linewidth from the experimental results, NumPy49,50 and SciPy51 were used to fit the Gaussian function to the PL spectrum in Fig. 2(a). Then, the PL linewidth is defined as the FWHM (Full Width at Half Maximum) of the PL spectrum, which is also the standard deviation of the fitted Gaussian function. The temperature dependent FWHM of the PL spectrum is shown as a solid circle in Fig. 4.

FIG. 4.

Temperature dependent FWHM of the PL peak in CsPbBr3. The blue solid circles are the experimental results, and the orange asterisks are the theoretical calculation results using Fermi’s golden rule. The theoretical broadening is obtained as the sum of 2lm(Σ) at the valence and conduction band in the case of Fermi’s golden rule, rigidly shifted by 44.7 meV to account for inhomogeneous broadening.

FIG. 4.

Temperature dependent FWHM of the PL peak in CsPbBr3. The blue solid circles are the experimental results, and the orange asterisks are the theoretical calculation results using Fermi’s golden rule. The theoretical broadening is obtained as the sum of 2lm(Σ) at the valence and conduction band in the case of Fermi’s golden rule, rigidly shifted by 44.7 meV to account for inhomogeneous broadening.

Close modal

The theoretical linewidth is calculated as the sum of the imaginary part of the electron–phonon self-energy at the valence band maximum (VBM) and the conduction band minimum (CBM). In order to account for the inhomogeneous broadening of the PL spectrum caused by disorder and imperfection,52 all theoretical linewidths are rigidly shifted by 44.7 meV and shown as asterisks in Fig. 4. Comparing the theoretical results with the experimental results, both results are consistent in the range of 100 K–400 K. The temperature dependence linewidth of a semiconductor can be expressed as53 

(4)

where γ0 is the inhomogeneous term, γAC is the electron–phonon coupling due to acoustic phonons, γLO is the electron–phonon coupling due to LO phonons, and ELO is relative to LO phonon energy. Equation (4) clearly shows that the PL spectrum broadening is dominated by the LO phonons. At first, Eq. (4) is used to fit the theoretical results; then, we can obtain γAC = 3.2 × 10−2 meV/K, γLO = 41.7 meV, and ELO = 22.2 meV. At a room temperature of about 300 K, the linewidth due to AC phonons is about 9.6 meV, while the linewidth due to LO phonons is about 30.7 meV. Compared with the calculation phonon DOS shown in Fig. 1(a), our fitting result of ELO is slightly higher because most of the calculated phonon energy is less than 20 meV. Recently, Ramade et al. have measured the electron–phonon coupling in CsPbBr3 nanocrystals and determined that γAC = 8 × 10−3 meV/K, γLO = 42 meV, and ELO = 16 meV.54 Compared with the results of nanocrystals, we find that the γLO value we got is almost the same as that of nanocrystals, while the value of ELO is slightly larger. Only the measured value of γAC is about 4 times higher. However, the value of γAC is too small so that there can be a large fluctuation of the fitted value of γAC. Nevertheless, these results are comparable. Furthermore, around room temperature, Eq. (4) can be simply approximated to a linear function, which is defined as

(5)

where γ0 = γAC + γLOkB/ELO = 0.19 meV/K. These results clearly indicate that the LO phonon scattering dominates the PL spectrum broadening even at room temperature, which is consistent with the case of MAPbBr3citeWright2016.

In summary, we have studied the electron–phonon interaction of CsPbBr3 both theoretically and experimentally. The temperature dependent PL linewidth has been extracted from the experimental data, and the theoretical linewidth has been obtained from DFT and many-body perturbation theory calculation. These two results are consistent with each other. We demonstrate that (1) the electron–phonon interaction of CsPbBr3 is dominated by the interaction between the electron and LO phonon in the temperature range of 100 K–400 K and (2) the PL spectrum linewidth is approximately linear with the temperature in the observable temperature range. These results not only lay the foundation for the quantitative model of electron–phonon coupling of perovskite materials but also provide guidance for perovskite based optoelectronic devices.

All authors contributed equally to this work.

The authors acknowledge financial support from the NSFC (Grant No. 61704032) and the Open Subject of State Key Laboratory of Computer Architecture (Grant No. CARCH201814).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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