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Optical Properties of Organometallic Perovskite: An ab initio Study using Relativistic GW Correction and Bethe-Salpeter Equation

In the development of highly efficient photovoltaic cells, solid perovskite systems have demonstrated unprecedented promise, with the figure of merit exceeding nineteen percent of efficiency. In this paper, we investigate the optical and vibrational properties of organometallic cubic perovskite CH3NH3PbI3 using first-principles calculations. For accurate theoretical description, we go beyond conventional density functional theory (DFT), and calculated optical conductivity using relativist quasi-particle (GW) correction. Incorporating these many-body effects, we further solve Bethe-Salpeter equations (BSE) for excitons, and found enhanced optical conductivity near the gap edge. Due to the presence of organic methylammonium cations near the center of the perovskite cell, the system is sensitive to low energy vibrational modes. We estimate the phonon modes of CH3NH3PbI3 using small displacement approach, and further calculate the infrared absorption (IR) spectra. Qualitatively, our calculations of low-energy phonon frequencies are in good agreement with our terahertz measurements. Therefore, for both energy scales (around 2 eV and 0-20 meV), our calculations reveal the importance of many-body effects and their contributions to the desirable optical properties in the cubic organometallic perovskites system.

Introduction. -Current progress in hybrid perovskite systems have shown significant promise in developing efficient yet low cost photovoltaics [1]. Since the introduction of organometallic perovskite CH 3 NH 3 PbX 3 (X=Cl, Br, I) by Kojima et al. [2], there have been several experimental investigations on different phases of these systems. Higher power conversion efficiencies (PCE) have been reported by several groups, which exceeds 12% conversion rate [3,4]. Recently Heo et al. [5] have found 12.3% PCE in CH 3 NH 3 PbI 3 on TiO 2 substrate. The thin films of CH 3 NH 3 PbI 3 have also been reported [6][7][8] to have a very high diffusion (about 100 nm) length for both electrons and holes, which makes them excellent candidates for optoelectronic devices. This leads one to believe that excitons may play an important role for energy transfer mechanism in these systems. In addition, there have also been several experimental [9,10] and theoretical [11,12] studies on the structural stability in hybrid perovskite CH 3 NH 3 PbI 3 . The high sensitivity of fundamental band gap to various stable crystal-structure phases of CH 3 NH 3 PbI 3 has been reported [11,12], suggesting the importance of the structure stability in both solar-cell applications and optoelectronic devices. More recently, it has also been predicted [13] that the cation-induced structure variability can promote strikingly different electronic and optical properties.
In this Letter, we focus on the excitonic effects on the optical conductivity of cubic CH 3 NH 3 PbI 3 using firstprinciples methods. The excitons are electron-hole pairs, a phenomenon which is often found in wide band-gap semi-  The k-paths along high symmetry points, along which the band structure is displayed.
conducting systems. These are associated with bound states formed between the excited electrons and the remaining holes. To capture these excitonic effects, more specifically the Coulomb interaction between the excited electrons and the remaining holes, we go beyond the random-phase approximation (RPA) and DFT-based single particle theory, and solve the equation of motion for two-particle response functions, formally known as Bethe-Salpeter equations (BSE). [14][15][16] Due to the presence of significant spin-orbit (SO) coupling in Pb ions, we also perform scalar-relativistic calculation and compare the results with non-relativistic (no SO coupling) calculations. We observe that both spin-orbit coupling and BSE with GW-correction play a significant role in optical properties of CH 3 NH 3 PbI 3 system. We find that the obtained exciton has about 0.3 eV of binding energy for a band gap of 1.9 eV. Our results may provide an alternative interpretation of optical response as measured by photoluminescence spectroscopy [17]. We also calculate the vibrational modes and infrared absorption (IR) spectra based on the ionically relaxed structure of cubic CH 3 NH 3 PbI 3 . The modes in the low frequency region are in a qualitative agreement with dominant peak features observed in our THz conductivity measurement on a CH 3 NH 3 PbI 3 thin film.

Computational Details and Experimental
Method. -Experimental characterization of the crystal structure of hybrid perovskites are difficult. The X-ray diffraction on high-quality CH 3 NH 3 PbI 3 crystals has identified cubic, tetragonal, and orthorhombic phases; while the transmission electron microscopy has suggested a pseudo-cubic phase to be consistent with the variability in the octahedral tilting [18]. Throughout the paper, the cubic phase is considered by neglecting small distortions as in Ref. [11]. We perform the first-principles calculations in several steps. We start with a DFT-based structural optimization by relaxing the internal coordinate of the ions in the cubic CH 3 NH 3 PbI 3 unit cells while keeping the lattice constant and cell shape fixed. The force on each atom was optimized within 5 meV/Å. For these calculations, we used fixed value of a = b = c = 6.26Å for the lattice constants which are obtained from the powder diffraction experiment [11,18].
For the above DFT step, we use ab initio package VASP [19] and choose ultra-soft pseudo potentials [20,21] with PBE exchange correlation functional [22,23]. The first Brillouin zone (BZ) is sampled with 4 × 4 × 4 kpoints using the Monkhorst Pack grid. Once the ionic and electronic relaxations have been achieved, we employ the small-displacement approach to calculate the vibrational eigen-frequencies and eigenvectors. Linear response method is used to calculate the Born effective charge tensors. The infrared-active modes are then obtained by the corresponding oscillator strength [24]: where Z * αβ (i) is the effective charge tensor of i-th atom, v β (i|ω n ) is the vibrational eigenvector for i-th atom and nth eigen-frequency ω n , and α, β are the three components of the cartesian axis. Using these oscillator strengths, we simulate the experimentally observable IR spectra based on a simple analytical expression [25]: where δ m (ω) are delta like functions defined by : The broadening of the Lorentzian peaks are adjusted to have the best agreement with experimental data (see Fig. 4). For the excitonic contribution to the optical conductivity, we perform a single shot GW correction on top of our DFT calculations, and finally solve the BSE as implemented in the code YAMBO [26]. In Σ(ω) = G 0 (ω)W (ω) self-energy formalism, G 0 stands for the non-interacting Green's function, where W is the screened Coulomb interaction, W (r, r , ω) = −1 (r, r , ω)V(r, r ). Here, V (r, r ) is the bare Coulomb interaction. The frequency dependent dielectric function (r, r , ω) is calculated from the response function χ(ω) as [27][28][29][30]: where within the random phase approximation (RPA) [14,27], the response function χ G,G (q, ω) can be calculated from non-interacting χ 0 and Green's function G 0 .  The quantity q = k − k is momentum transfer and G is the reciprocal lattice vectors. This method was first proposed by Lars Hedin [31], and further details can be found in Ref. [28,[32][33][34][35]. It has been found that the inclusion of spin-orbit coupling in the GW calculations is also important in 5f -electron materials [36,37]. This GW method is a many-body perturbation (MBPT) technique which accounts for missing dynamical correlation in the DFT. This GW-correction method has been used to successfully predict accurate band gaps for various narrow-gap semiconductors. Therefore, in CH 3 NH 3 PbI 3 system under consideration, we perform GW correction on top of the DFT calculations.
However, for systems with larger band gap, the RPA is no longer adequate, and one has to incorporate electronhole (e-h) correlation in the response function χ. This is the Bethe-Salpeter equation (BSE), where one calculates the response function in terms of the noninteracting two particle e-h Green's function. The details of this formalism can be found in Ref. [14,26]. Alternatively, one can also solve the BSE by reducing the problem to a two-particle e-h Hamiltonian [14]. The eigenvectors of this Hamiltonian are the excitonic states |λ with E λ being the corresponding excitonic binding energies within the Tamm-Dancoff [38] approximation, in which only positive energies are considered after diagonalizing the non-Hermitian e-h Hamiltonian [26]. For narrow-gap semiconductors, the effect of BS is negligible; while for wide gap semiconductors, E λ are often within the gap and can thus have significant contribution to the optical properties. The YAMBO program uses single-particle quantities based on the DFT calculations within the Quantum Espresso (QE) [39]. Therefore, we export the VASP optimized structure to QE and, through the ionic force calculation, verify the optimized structure from the VASP. The relaxed structure is then used to calculate the electronic band-structure and total density of states. We use Normconserving PBE based pseudopotentials in QE. To account for SO coupling we we use the scalar-relativistic PBE pseudopotential. For both cases (with and without SO coupling), QE generates DFT band-structure, which are then renormalized with GW corrections using YAMBO. Finally, excitonic contributions to the optical properties are incorporated on top of DFT and DFT+GW calculations including SO coupling.
The THz transmission spectra of the CH 3 NH 3 PbI 3 perovskite sample was obtained using Teraview TPS3000 THz time-domain spectrometer, coupled with a Janis ST-100-FTIR cryostat. The perovskite film of thickness 230 nm is deposited on a z-cut quartz substrate. A bare substrate is used for the reference run. By fitting the spectra with the thin-film-on-substrate transmission expression, one obtains the complex THz conductivity of the sample. The experimental details are described in Ref. [40]. Data taken at 300 K and 20 K are presented in the present work.
Results. -All our calculations presented in the Letter are based on highly optimized structure of cubic CH 3 NH 3 PbI 3 system. The ionic relaxation are performed for internal coordinates while we kept the lattice parameters fixed. The final configurations of the atoms are shown in Fig. 1(a). In the lower panel of Fig. 1(a), we show x, y, and z projected cross-sectional view, which shows the final orientation of methylammonium cation inside the cubic cell. For the band structure calculations, we use high symmetry path inside the first Brillouin zone as shown in Fig. 1(b). SO coupling and QP correction. We have systematically studied the effect of SO coupling as well as GW self-energy corrections on the fundamental band gap of cubic CH 3 NH 3 PbI 3 . The PBE-DFT calculations without SO predicts 1.5 eV gap (solid red line in Fig. 2(a)) while GW correction (without SO) estimates the gap to be 2.5 eV (solid blue line in Fig. 2(a)). The inclusion of SO coupling reduced the band gap to be 1.4 eV for DFT (solid red p-3 line in Fig. 2(c)) while the GW corrections estimates the band gap to be 1.9 eV (solid blue curve in Fig. 2(c)). The corresponding total density of states (DOS) are shown in Fig. 2(b) and (d). The experimental band gap for this system was found to be 1.61 eV at room temperature. [17] Therefore, we see that the inclusion of SO coupling underestimates the band gap while the GW correction without SO overestimates the gap by 0.9 eV. Our systematic study reveals that the best agreement with experiment is reached when we include both the SO coupling and GW self-energy correction. We notice that the the experimental value of the band gap is obtained at room temperature, where the thermal effect could reduce the band gap as compared to that in the ground state. All our calculated values of band gap at various conditions are listed in Table I. We point out that our finding of the overall effects of self-energy and SO coupling on band gap is consistent with earlier results reported by Brivio et al. [41].
Excitonic Effect. In the presence of SO coupling, we calculated the dielectric function and optical conductivity by solving BSE with (Fig. 3(b)) and without ( Fig. 3(a)) the self-energy correction. The peaks in optical conductivity represent the electronic excitations from valence band to conduction band satisfying the optical selection rule [42,43]. Therefore, the fundamental gap can be represented by the location of the first peak in optical conductivity spectrum. By incorporating the excitons through the solutions of BSE, we find enhanced peak appearing below the gap edge. These peaks are indicated by the arrows in Fig. 3. We also find that the BSE solutions including GW self-energy correction, as shown in Fig. 3(b), give rise to more shifted excitonic contribution than without GW correction ( Fig. 3(a)). By comparing the location of this shifted excitonic peak with the band gap edge, we are able to deduce that the exciton binding energy is about 0.3 eV. Interestingly, the peak position due to the excitonic contribution is more close to the experimental value of 1.61 eV.
Vibrational Modes vs. Measurements. The presence of flexible methylammonium cation at the center of the rigid perovskite cage leads us to investigate the the vibrational modes, which can be probed by low-frequency spectroscopy such as Raman, IR, or THz conductivity. For this purpose, the vibrational mode energy and oscillator strength are calculated. The IR spectra are simulated with a Lorentzian fitting. The results are shown in Fig. 4, with the vertical blue lines denoting the positions of vibrational mode frequencies. We see that our calculated IR absorption spectra, as represented by the solid black curve, reproduces qualitatively similar peak features observed in experimental data for the THz conductivity (red and green lines), in the low frequency region (0-3 THz). The computed spectra is slightly stretched in the frequency range. Such stretch has also been reported by earlier ab initio calculations of Raman spectra for CH 3 NH 3 PbI 3 [44]. Our THz conductivity measurement demonstrates the possible energy and charge transfer mechanism in cubic CH 3 NH 3 PbI 3 through the low-energy vibrational phonon modes.
Conclusion. -In summary, we have demonstrated the effect of excitons and vibrational modes on the optical properties of cubic organometallic perovskite CH 3 NH 3 PbI 3 . All our calculations have been based on internally relaxed stable ionic structure with the experimen-tal lattice parameters. Our study has uncovered the importance of both the spin-orbit coupling and quasi-particle self-energy correction, which conspire to reproduce the experimentally observed fundamental band gap. In this wide-gap semiconductor, we have found significant excitonic contribution to the optical conductivity at ∼ 1.5 -2.0 eV using first-principles BSE calculations. In addition, we have also performed first-principles simulations of the vibrational modes due to the presence of flexible methylammonium cations at the center of the cubic perovskite cage. We have shown that the signature of the modes can be revealed in the IR spectra, with the profile very similar to the THz conductivity we have measured on a CH 3 NH 3 PbI 3 thin film. Therefore, our firstprinciples calculations have predicted that the features of the excitation spectra must originate from the strong spinorbit coupling, many-body effects, and low-energy vibrational modes. Our systematic first-principles study, for the first time, has benchmarked the various competing effects, which must be taken into account to understand and utilize the optoelectronic properties of CH 3 NH 3 PbI 3 to enhance its solar-cell applications.