Electronic and optical properties of bulk and surface of CsPbBr3 inorganic halide perovskite a first principles DFT 1/2 approach

This work aims to test the effectiveness of newly developed DFT-1/2 functional in calculating the electronic and optical properties of inorganic lead halide perovskites CsPbBr3. Herein, from DFT-1/2 we have obtained the direct band gap of 2.36 eV and 3.82 eV for orthorhombic bulk and 001-surface, respectively. The calculated energy band gap is in qualitative agreement with the experimental findings. The bandgap of ultra-thin film of CsPbBr3 is found to be 3.82 eV, which is more than the expected range 1.23-3.10 eV. However, we have found that the bandgap can be reduced by increasing the surface thickness. Thus, the system under investigation looks promising for optoelectronic and photocatalysis applications, due to the bandgap matching and high optical absorption in UV–Vis (Ultra violet and visible spectrum) range of electro-magnetic(em) radiation.

www.nature.com/scientificreports/ Additionally, solid-state inorganic metal-halide perovskite materials are promising with high stability at ambient condition 20-23 as compared to organic-inorganic hybrid halide perovskites. The solid-state inorganic single perovskite materials have the chemical formula of AMX 3 , where A is a cation (Li, Na, K, Rb, Cs, etc.), M is another cation (typically Pb/Sn) and X is a halide (F, Cl, Br, I). Almost, all-inorganic halide perovskites such as CsPbBr 3 , CsPbI 3 , and their alloys [CsPbBr x I 3−x ] are shown to have greater stability as compared to CH 3 NH 3 PbI 3 , while their optoelectronic properties are in line with organic-inorganic hybrids (CH 3 NH 3 PbI 3 ) 24,25 . The long-term stability of inorganic solid-state perovskite materials are the primary impediment to their widespread implementation. A single inorganic perovskite compounds APbBr 3 (where A is Li, Na, K, Rb, and Cs) display semiconducting characteristics with energy bandgap in the range of 1.708-1.769 eV 6 . Recently, McGrath et al., synthesised highly stable nanocrystals of the inorganic halide perovskites CsPbBr 3 employing oleylamine/ alkylphosphonic acid displaying outstanding photo-physical and chemical properties 26 . At the experimental level, perovskite/silicon solar cells are in tandem with enhanced efficiencies when compared with other perovskite-based multi-junctions. Beal et al., reported an improved chemical-stability and efficiency up to 10.77% for tandem solar cells based on solid state inorganic CsPbI 3 perovskite 27 . Ouedraogo et al. 28 , have suggested the different strategies and fabrication method to improve the stability of black phase of CsPbI 3 such as Solventadditives engineering, Alloying/element doping engineering, 2D nanocrystal engineering. On the other hand, calorimetric investigations on the formation enthalpies of CsPbX 3 perovskites have shown the diminishing order of thermodynamic stability for CsPbCl 3 , CsPbBr 3 , and CsPbI 3 29 . From various studies it has been reported that CsPbBr 3 undergoes structural phase transitions from orthorhombic to tetragonal and subsequently to cubic at higher temperatures Apart from their superior bulk properties, it is highly crucial to preserve such intriguing properties at their surfaces and interfaces for device applications. Surfaces and interfaces play a key role in device fabrications and deciding the device's performance. Thus it needs the critical understanding and thorough knowledge of the surface characteristics of inorganic perovskites. A substantial influence of surface phenomena (surface energy, atomic structures, and electronic structures, etc.) on material stability and device performance has been seen in nanostructures due to their high surface-to-volume ratio. The influence of surface energy on the stability of CsPbX 3 (X = Cl, Br, I) nanocrystals 37 and stability of meta-stable nanocrystal via strain effect have been reported in several studies [38][39][40] . The surface-guided growth techniques have been used to synthesise the CsPbBr 3 nanowires 41 . The experimental study has identified that 2D CsPbBr 3 with CsBr-terminated (100) surface is the most stable one 41,42 . Theoretically, the first principles study using VASP (Vienna Ab initio Simulation Package) has reconfirmed the ground state stability of non-polar CsBr-terminated (100) surface as compared to (110), and (111) polar-surfaces 43 .
A first principles DFT 44 is used to comprehend the experimental findings and gain insight into atomisticscale interactions in deriving the varied physical properties of a material. Ghaithan et al., performed a fist principles DFT calculation on CsPbBr 3−x Cl x Perovskite using PBE-GGA and mBJ-GGA functional 45 . They have reported an enhanced band gap from 2.23 to 2.90 eV with increasing Cl concentration within the modified Becke-Johnson generalized gradient approximation (mBJ-GGA) potential 45,46 Here, in this study we concentrate on the bulk orthorhombic phase of CsPbBr 3 which exist below 80 °C may be favourable for ground state DFT based first-principles calculation. We have also performed the ground state calculation on the surface stability of orthorhombic CsPbBr 3 .

Computational details
For computation of both the bulk and surface of CsPbBr 3 we have used Kohn-Sham DFT (KS-DFT) based Atomistic Simulation Software QuantumATK (VNL-ATK) which incorporate Linear Combination of Atomic Orbital (LCAO) basis function 47,48 . All electrons are treated by a newly modified potential ( V S ) that has been developed by correcting the self-interaction error within the exchange-correlation functional via semi-empirical approach often denoted as DFT-1/2 49,50 . In this approach the general effective potential V eff in KS-DFT equation has been substituted to obtain a modified potential V mod =V eff − V S . The corrected term V S is analogous to the half-occupation state of Slater scheme, mimics the electrostatic potential of the atoms in the crystal. Since it is impossible to sum all the divergent potentials for infinite periodic lattices, the extended part is truncated using the step function.
Here, cut off radius r cut estimated variationally, n = 8 and A is the amplitude of the correction. The corrected term �(r)V S (r) is confined within a sphere of radius r cut , enable its application in band-structure calculation. This means DFT-1/2 is plausible for improving the degenerated semiconductor bandgap up to 10 eV 49,50 . Tao et al. has reported the performance of DFT-1/2 in opening the bandgap in metal halide perovskites, comparable to that of GW 51 . Moreover, we are familiar with DFT-1/2 approach and reported increased band gap in our previous studies of 1D (6,1) single walled Carbon nanotube 52 and 2D hexagonal ZnSe 53 . At low temperature CsPbBr 3 crystallizes in orthorhombic phase having space group Pnma. An ultra-thin 001-surface structure has been cleaved using 2 × 2 × 1 supercell with the repetition of unit cell along x and y direction. A vacuum of 15 (Å) is imposed along the z-axis to interrupt the lattice periodicity which eventually avoid nonphysical interaction of the wave functions. We have considered the bulk and surface of orthorhombic CsPbBr 3 and structural www.nature.com/scientificreports/ optimization was performed from PBE-GGA 54 , rather than DFT-1/2 due to its limitation in calculating the total energy. The orthorhombic crystal of CsPbBr 3 and 001 surface are presented in Fig. 1a-d.

Result and discussion
The energy-volume curve is obtained by fitting the data obtained from first principles self-consistent force (SCF) calculation in the third order Murnaghan equation of state 55 as The volume optimization curve for both the bulk and the surface is shown in Fig. 2a and b. The atomic positions, previous experimental/theoretical lattice constants 45,56 and optimized lattice constants are presented in Table 1. Our calculated lattice constant varies by a very small amount,  45 , 21.56 57 and 18 GPa 56 . We also report the bulk modulus of surface CsPbBr 3 ∼ 9.618 GPa which is almost half of its bulk counterpart.
Electronic properties. The optimized structure has been used for the calculation of electronic and optical properties of both the bulk and surface structure from DFT-1/2 approach. The importance of DFT-1/2 in predicting accurate electronic bandgap ( ≤ 10 eV) and convergence feasibility has already been discussed. In our study both bulk and surface structure exhibit a semiconducting behaviour with a bandgap of ∼ 2.36 eV and ∼ 3.82 eV, respectively (see Figs. 3, 4). The presence of Fermi level (E F ) at the middle of the bandgap indicate its intrinsic behaviour with a direct bandgap along Ŵ − Ŵ symmetry. Our result of electronic band-gaps calculated from semi-local DFT-1/2 are in consistent with the previous results obtained from higher order DFT functional like GW, HSE, mBJ etc., and experiment [for numerical comparison see Table 2]. Meanwhile, DFT-1/2 open up the the underestimated bandgap from GGA and local density approximation (LDA) by ∼ 20%. Our result of bandgap (2.36 eV) for orthorhombic CsPbBr 3 from DFT-1/2 approximation is in consistent with the experimental value of 2.23 eV 58 , 2.446 eV 59 and 2.36 eV 60 . It also agrees well with the result (2.28 eV) of higher order   www.nature.com/scientificreports/ morphology has been preserved with its direct band gap along Ŵ − Ŵ symmetry. Similar to the bulk, we have noticed a coupling between Pb-Br, and Cs-Br ions, suggesting covalent and ionic bond, respectively, in surface as well (see Fig. 4). Brescia et al. 59 , formulated an expression for theoretical prediction of onset band gap by fitting a power function with a set of experimentally determined bandgap as a function of film thickness ′ t ′ in relation to effective mass m * .
where E bulk g and E t g are the energy band gap of bulk and film of thickness (t) respectively. It is obvious from the above expression that the decrease in film thickness will increase the energy band gap of the thin film. In Fig. 6, we have presented the variation of energy bandgap as a funtion of slab thickness of CsPbBr 3 . The curve is fitted based on the Eq. (4) using gnuplot. The similar trend of increase in band gap from 2.47 eV to 2.84 eV on decreasing the slab from 15→ 1 layer has also been reported from PWscf PBE-SOC (spin-orbit coupling) calculation 70 .  www.nature.com/scientificreports/ The increased in the direct band gap of the surface state may be infer to quantum confinement effect 74 . The halide based nanoparticles perovskite exhibit quantum confinement effect for a nanometer-size (< 50 nm) and strong Mie resonances above 10 2 nm 75 . We have arbitrarily determine the electron effective mass of 001-surface using Eq. (4), taking the single layer thickness t = 8.398 Å, E t g = 3.82 eV and E bulk g = 2.36 eV. The estimated electron effective mass of 001-surface is found to be ∼0.20m e , in good agreement with 0.24m e (single slab) and 0.17m e (bulk) 70 .
Optical properties. For better explanation of the optical response the electron transitions are considered from the first three valence bands below E F to first three conduction bands above the E F . The probability transition along the various symmetry points in the first Brillouin zone and their corresponding maximum energies give the information about the optical properties. The optical response with respect to the incident photon energy is presented in the form of a complex dielectric function given by;  The absorption coefficients α(ω) , which is related to the dielectric function is given as follows; The real part of refractive index (n) is given by The optical parameters of both bulk and surface like real part of dielectric function ( ε 1 (ω) ), imaginary part of dielectric function(ε 2 (ω) ), absorption coefficient(α(ω) , reflectivity(r(ω) ) and refractive index ( n(ω) ) are presented in Figs. 7, 8 and 9. In all figures we have used black, red and green color lines to represent the polarization along x, y and z-axes, respectively. Fig.7a and b represent the real and imaginary part of dielectric function for the bulk system. The ε 1 (ω) rises slowly and reach maximum at ∼2.35 eV, in which the dielectric polarization along z-axis predominate. After 2.5 eV the ε 1 (ω) decreases sharply and drops below 0 at ∼3.0 eV. The drop of ε 1 (ω) below zero gives negative value in which the incident photon beam is attenuated due to the dissipation of energy into the medium and giving rise to metallic behaviour. The drop of ε 1 (ω) from maximum value can be relate to the rise of ε 2 (ω) , suggesting inter-band transition. The intensity of ε 1 (ω) increases slowly after 3.0 eV and gives a constant intensity of ∼1.0 for all values of photon energy above 5.0 eV. The static value of real dielectric constant [ ε 1 (0) ] is inversely related to the energy bandgap E g by; From the above Eq. (10), the higher value of bandgap leads to the low value of ε 1 (0) . It is well verified that the higher value of bandgap (3.82 eV) in the surface system gives far lower value of ε 1 (0)=1.10, as compared to the bulk system [ ε 1 (0)=1.72]. The calculated values of ε 1 (0) along the different polarization direction is tabulated in Table 3. In ε 2 (ω) spectra the maximum peak occurs at around 3.45 eV (see Fig. 7b). We have presented the Figure 6. The variation of energy bandgap (E g ) as a function of slab thickness (t) of 001-surface of CrPbBr 3 using DFT-1/2 (blue line represent the results of energy bandgap and red dot denotes the data fit using Eq. (4) from gnuplot). www.nature.com/scientificreports/ ε 2 (ω) spectra of 001 surface of CsPbBr 3 obtained from DFT-1/2 in Fig.7d. Unfortunately, we did not find any experimental result of ε 2 (ω) spectra of the thin film of CsPbBr 3 for direct comparision. Therefore, the calculated ε 2 (ω) is compared with the available result of nanocrystalline orthorhombic CsPbBr 3 calculated at different temperatures from high-resolution spectroscopic ellipsometry, high-resolution transmission electron microscopy and terahertz spectroscopy measurements 76 . The theoretical ε 2 (ω) calculated from DFT-1/2 and experimental 76 ε 2 (ω) are plotted together in Fig. 7d for better comparision. Here, ε 1 (ω) along x and y-axes predominates, due to the 2D nature of CsPbBr 3 . The maximum value of ε 1 (ω) occurs at ∼4.20 eV. It seems that the maximum peak of ε 1 (ω) spectra has shifted towards the higher energy as compared to its bulk counter part. The highest peak of ε 2 (ω) is located at ∼4.60 eV. The absorption coefficient ( α ) for both the bulk and the surface with respect to the photon energy and the wavelength is displayed in Fig. 8a-d. The positive tangent drawn on the α(ω) cut the x-axis somewhere at ∼2.30  www.nature.com/scientificreports/ eV, which may be considered as an optical band gap (Fig. 8a). From, Fig. 8b, the estimated absorption onset is ∼ 540 nm, which is in good agreement with 554 nm 58 . Majhi et al., prepared the orthorhombic CsPbBr 3 from wet chemical synthesis method and studied the optoelectronic properties by UV-Vis and photoluminescence (PL) spectroscopy in which they reported the band-to-band transition at 554 nm (2.23 eV) 58 , which may be considered as an optical band gap. This spectra can be related to the first electron transition from top of the valence band to the bottom of conduction band along Ŵ − Ŵ direction. For the bulk system an absorption peak of magnitude ∼4.5×10 5 cm −1 appears at ∼4.0 eV, has zero absorption intensity in the range of 0-2 eV and beyond 6.0 eV (see Fig. 8a). This proves that the for the bulk CsPbBr 3 the energy active window lies between 2.3 and 5.0 eV, having width of ∼2.70 eV. We have observed a reduce absorption peak in the case of surface system. The intensity of the peak for the surface is ∼1.58×10 5 cm −1 at ∼5.60 eV (see Fig.8c). The magnitude of α = ∼1.58×10 5 cm −1 comprehended well with the experimental result 77 . The energy active window found in between 4 and 6.5 eV, giving rise to the window width of ∼2.5 eV. The window width for the bulk system is little high as compared to the surface by W=0.20 eV. The absorption onset is found at ∼340 nm (see Fig.8d). There is a shift in the absorption peak by ∼1.60 eV towards the higher energy while scaling from the infinite layers (bulk) to a single layer (surface). We have observed a prominent feature of a blue-shift in diminishing a structure size from bulk to surface(ultra thin film). It has already been shown that the perovskite with the slab thickness below 3.0 nm exhibit a blue shift [see Fig. 3 of Ref. 70 ]. Most of the MAPbBr 3 perovskite nanoparticles of larger sizes (more than 50-100 nm) exhibit the influence of enhanced Mie modes preserving the blue shift despite the of feeble quantum confinement effect 75 . Electron energy-loss spectroscopy (EELS) reported the onset optical gap of around 2.4-2.5 eV for a 10 nm thick thin film of CsPbBr 3 59 . The thin film of CsPbBr 3 perovskite prepared from Single Source Thermal Ablation shows absorption onset at 530 nm 78 . Also, the CsPbBr 3 film prepared by 2-step sequential deposition, exhibit prominent absorbance peaks at 510-525 nm, with absorbance onset at 532 nm 79 .
We have presented the calculated spectra of reflectivity and the refractive index for both the bulk and surface systems in Fig. 9a-d. At the initial part of the photon energy, the reflectivity spectra is very low and increases with the increase in the energy and ultimately give maximum reflectivity of ∼35.0% at ∼4.0 eV (see Fig. 9a). Meanwhile, the maximum reflectivity of the surface is only ∼2.0%, this may be due to its transparent behaviour within UV-Vis range of the electro-magnetic radiation Fig. 9c. The maximum surface reflectivity occurs at ∼5.60  Table 3. Static real part of dielectric function ε 1 (0)and static refractive index n(0) of orthorhombic bulk and surface of CsPbBr 3 .  Fig. 9b and d, shows the refractive index of the bulk and the surface. The static refractive indices for the bulk system along x and z-direction are 1.72 and 1.80, respectively. This result is in good agreement with the previous result of 1.96 45,46 . From the static value, the n(ω) increases to reach the maximum value at ∼3.20 eV. Beyond 3.20 eV the spectra of n(ω) decreases rapidly and goes below 1.0 at 3.40 eV in the UV-region. If we refer to Fig. 7a, we can see the value of ε 1 (ω) becomes negative at this value of energy ( ∼3.40 eV). The presence of n(ω) value below 1.0 is nonphysical in which the phase velocity move faster than the group velocity (speed of light). This can be related to the occurrence of the plasmonic vibration with the plasmonic frequency ( ω p ) close to resonance frequency. The static refractive index of the surface is found to be n(0)=1.045 which is close to 1.0, indicating transparent behaviour within visible range of light. In case of the surface the plasmonic vibration lies at ∼5.2 eV.

Conclusions
For the first time, we have carried out the electronic and optical properties of the orthorhombic bulk and 001-surface of CsPbBr 3 from DFT-1/2 approach. This work also performed to test the efficiency of DFT-1/2 in deriving the electronic and optical properties of inorganic perovskite. Herein, we report the efficiency of DFT-1/2 is as effective as that of higher order DFT like HSE hybrid functional in calculating the electronic bandgap. The calculated band gap is 2.36 eV in good agreement with the experiment. While, the ultra-thin surface slab of CsPbBr 3 open up the electronic bandgap by ∼42%. We report the decrease in bandgap while increasing the slab thickness of CsPbBr 3 film. The thin film of CsPbBr 3 exhibit tunability of the bandgap via the surface thickness modification. The presence of high value of absorption coefficient ∼4.5×10 5 cm −1 and ∼1.58×10 5 cm −1 in UV-Vis energy range for both the bulk and the surface, respectively. The tunability of energy bandgap offers remarkable optoelectronic properties in UV-Vis range making this material promising for optoelectronic applications.