Role of an external electric field on hybrid halide perovskite CH3NH3PbI3 band gaps

The organic-inorganic perovskite CH3NH3PbI3 has attracted much attention due to their power conversion efficiency as a potential photovoltaic material, but the role of an external electric field has not been well understood. Based on first-principles calculations, the effects of an external electric field (E) applied along the [111] direction of the orthorhombic perovskite, CH3NH3PbI3, on its electronic structure and optical properties are investigated. Our results indicate that the electric field strength affects the band gap (Eg) of CH3NH3PbI3 (MAPbI3, MA = CH3NH3). The energy difference between the two peaks closest to the Fermi level in the density of states diagram decreases with increasing applied electric field strength along the [111] direction, indicating that the covalent character increases between A-sites cations and I-sites anions. Both the cell volume and the final energy show the same increasing trend. The absorption peaks move toward the visible-frequency range, with the optimal band gap of 1.1–1.45 eV and E = 0.04–0.06 eV/Å/e. In particular, the non-linear change of the second-order Stark effect causes a non-linear change in the band gap.

YMnO 3 , and BiFeO 3 . Bellaiche et al. reported that an external electric field could induce polarization paths in PbZr 1−x Ti x O 3 perovskites and lead to the expected sequence of tetragonal, A-type monoclinic, and rhombohedral structures 19 . Xu et al. 20 demonstrated that electric fields could induce a change from a ferroelectric phase to an antiferroelectric phase in a lead-free NaNbO 3 -based polycrystalline ceramic. Therefore, it is very important and valuable to investigate the relationship between external electric fields and the physical properties of the organic-inorganic hybrid perovskite CH 3 NH 3 PbI 3 . CH 3 NH 3 PbI 3 undergoes two phase transitions, one at 160 K (orthorhombic to tetragonal) and the other at 330 K (tetragonal to cubic) 21 . Using Density functional theory (DFT) calculations, Leppert et al. investigated the Rashba effect induced by the electric field and strained in the hybrid halide perovskite CH 3 NH 3 PbI 3 with a tetragonal and cubic structure 22 . Although the orthorhombic phase is realized with rotations about the C-N axis that freeze out when T ≤ 162 K, temperature is not the only factor affecting these properties. An external electric field or magnetic field could also be an important factor, so we chose the orthorhombic CH 3 NH 3 PbI 3 as our subject to investigate these effects.
In this article, we investigate the effects of an external electric field applied along the [111] direction on the geometry structure, electronic energy band structure, total density of states, and optical properties of CH 3 NH 3 PbI 3 . This study provides a method for obtaining the optimal band gap of CH 3 NH 3 PbI 3 and expands the scope of its applications.

Results and Discussion
The effects of the external electric field (E) direction, including the [001], [010], [100], [110], and [111] directions, on the band structure of CH 3 NH 3 PbI 3 were studied. The application of an electric field along the [111] direction in CH 3 NH 3 PbI 3 decreases the band gap, while fields aligned along the other directions increase the band gap. Therefore, we only investigated the physical properties of CH 3 NH 3 PbI 3 under an electric field aligned along the [111] direction to obtain the optimal band gap of 1.1-1.4 eV.
Band structure of CH 3 NH 3 PbI 3 under the external electric field. The band structure of CH 3 NH 3 PbI 3 in the absence of an external electric field is shown in Fig. 1 (a,b). When the external electric field is equal to zero, the valence band maximum (VBM) and the conduction band minimum (CBM) are located at the same Γ-point, which indicates that CH 3 NH 3 PbI 3 possesses direct semiconductor characteristics. The calculated band is 1.726 eV and 1.675 eV, corresponding to the generalized gradient approximation (GGA) functional developed by Perdew, Burke, and Ernzerhof PBE 23   of conduction bands calculated using GGA + PBEsol is greater than that calculated using GGA + PBE. In the presence of an applied external electric field with a strength ranging from 0.01 to 0.06 eV/Å/e, the band structures of the orthorhombic CH 3 NH 3 PbI 3 are similar. Figure 1 (c,d) show the band structure when E = 0.06 eV/Å/e. The conduction band shifts downward to the Fermi level (0 eV) and the energy band near the CBM is more dispersed, resulting in a change in the band gaps. All the configurations indicate an indirect band gap semiconductor, and the VBM and CBM are located at the centre of the Γ-point in k space. Because of the narrower indirect band gap of the CH 3 NH 3 PbI 3 semiconductor, only a small amount of energy is required for the formation of excitons. However, the GGA + PBE and GGA + PBEsol methods underestimate the band gap because of self-interaction errors, suggesting that the actual band gap of CH 3 NH 3 PbI 3 is slightly smaller than the calculated values.
The band gaps of CH 3 NH 3 PbI 3 are strongly affected by the applied electrical field strength. As the external electric field strength increases, the band gaps calculated by GGA + PBE first decrease linearly, then increase slightly, and finally, decrease linearly. The band gaps calculated by GGA + PBEsol indicated the optimal value between 1.1 and 1.4 eV for all the structures under an external electric field, as shown in Fig. 2. Therefore, both the results indicated that the external electric field contributes to a decrease in the band gap.
Density of states of CH 3 NH 3 PbI 3 under an external electric field. Figure 3 Figure 5 shows the partial density of states (PDOS) of CH 3 NH 3 PbI 3 (GGA + PBEsol) under applied external electric field strengths of E = 0.00 and 0.06 eV/Å/e. The electronic orbitals 5d6s6p, 5s5p, 2s2p, 2s2p, and 1s are modeled as the valence orbitals for Pb, I, C, N, and H, respectively. The s-p hybrid level increases with external electric field increasing, and the effect on the conduction band is much more pronounced than that on the valance band. The peak near the orbital energy level at −5.0 eV splits into two peaks, which is attributed to the Stark effect of the s and p electrons, and one peak position shifts to the Fermi level. The external electric field causes the d electrons of Pb to shift to higher energies, but the d electrons do not affect the s and p electrons.
The Stark effect can induce the splitting of degenerate energy levels. The degree of the band splitting increases with the electric field, and a smaller band gap can be obtained. All peaks of the orbital energy level in the DOS curves broaden and shift to lower energies as E increases. The physical mechanisms underlying the change in the band structure of CH 3 NH 3 PbI 3 is ascribed to the C-N and Pb-I bond lengths, changes in the lattice structure, and charge transfer between the Pb and I atoms. The lattice distortion caused by the mutation of lattice parameters may lead to change in the microscopic electric structure, band structure, and density of states, similar to the results reported in ref. 17 .   > , and the deviations range from 2.31% to 27.86%, which are much greater than those for single inorganic perovskites 27 .
In addition, the orientation of the CH 3 NH 3 + ion can be seen in the orthogonal CH 3 NH 3 PbI 3 . Table 1 shows the Cartesian coordinates of the C and N atoms are displaced by 0.01 Å along the [111] direction, which is similar to a past report that Pb and apical I atoms are displaced by 0.1 Å and 0.01 Å along the [001] direction in P4mm CH 3 NH 3 PbI 3 (ref. 22 ). The Pb-N-C angle decreases with increasing external electric field, which consistents with the increase in the lattice parameters and cell volume.
Optical properties of CH 3 NH 3 PbI 3 under an external electric field. CH 3 NH 3 PbI 3 may show different preferential growth directions with different substrates, so obtaining the optical performance in this growth direction is an important physical problem. Let us take the [100] direction as the preferred growth direction as an example to illustrate this problem. Figure 8 (a) calculated by GGA + PBE and (b) calculated by GGA + PBEsol show the optical absorption spectrum with polarized light, where the polarization is along the [100] direction of CH 3 NH 3 PbI 3 in the presence of an external electric field. It is seen that the electric field significantly influences the optical absorption characteristics of CH 3 NH 3 PbI 3 . We assume the highest intensity absorption peak as the main absorption peak. The values 5.5 eV and 3.68 eV in the range of 0~5 eV for CH 3 NH 3 PbI 3 calculated by the GGA + PBE and GGA + PBEsol methods are the positions of the main absorption peak without the external electric field, which corresponds to absorbed light with the highest frequency. As the external electric field increases, the peak positions shift to lower frequencies, approaching the visible light region, and full width at half maximum (FWHM) of the absorption peaks decreases, which makes light absorption more effective and thus improves the photoelectric conversion. In addition, there are some absorption peaks at higher energies beyond the visible range, which do not play a major role in photoelectric conversion, but can become a candidate as optical detection device such as ultraviolet band.
As the first-order approximation, the band gap E g and wavelength should satisfy the following relation: We can obtain the related data based on eq. 1, as shown in Table 2. The following can be deduced: (i) The band gap decreases non-linearly with increasing external electric field. When E ≤ 0.03 eV/Å/e, ν and ν 100 decrease with an increase in the external electric field; when E ≥ 0.04 eV/Å/e, ν and ν 100 increase with external electric field increasing, except ν with 0.04 eV/Å/e based on the GGA + PBE. Here, ν is the frequency calculated by the band gap, ν 100 is the frequency along the [100] direction based on the Fig. 8. (ii) The band gap (Eg) based on the band structure is similar to E g ' based on the optical properties. Based on adiabatic approximation and single electron approximation, the relationship between absorption coefficient and photon energy can be expressed as, where a is the absorption coefficient, B is a fitting parameter, h is Planck's constant, v is frequency, and E g ′ is the band gap. According to the data in Fig. 8, E g ′ values based on the optic properties are shown in Table 2. Moreover, although the preferred growth direction may be not [100] in practice, we provided a method to obtain the optical properties of CH 3 NH 3 PbI 3 under an external electric field.
Why does the band gap decrease non-linearly? For CH 3 NH 3 PbI 3 under an external electric field, the Hamiltonian can be written as,ˆˆ=   Here, × − e 1 6 10 C 19 ; E is the external electric field; and θ is the angle between the direction of the electric field and the radius vector direction → r . ′ H is the perturbation. Based on the perturbation theory, the energy levels (E i ) will change to the initial values (E (0) i ) by an amount ∆E i .
(1) ( 2) The energy changes owing to the electric field, where <ψ i | is the initial state of the system, then we have, For second-order energy changes, the summation is over all possible states of the system, so, If E can be considered a constant over the perturbation volume, Table 1, it can be seen that the movements of the I − anions are coupled to the movements of the monovalent MA + cations and the rotation of the MA dipoles. This change in polarizability in the domains can influence the second-order Stark effect through the change in the dielectric constant owing to the change in the optical absorption spectrum 28 . Then, ΔE (2) may reflect in two possible first-order Stark effects, as shown in Fig. 9. If the value of ΔE (2) is positive, the band gap will widen; if the value of ΔE (2) is negative, the band gap will shrink. However, the second-order Stark effect cannot be larger than the first-order Stark effect, so CH 3 NH 3 PbI 3 without an external electric field has the maximum band gap. The non-linear extent of the second-order Stark effect cause a non-linear change in the band gap.
The device required that: (i) The Highest Occupied Molecular Orbital (HOMO) of TiO 2 as ETL layer must be lower than CBM of perovskite active layer. (ii) The Lowest Unoccupied Molecular Orbital (LUMO) of spiro-omeTAD as HTL layer must be higher than VBM of perovskite active layer. The external electric field induced the Stark effect, splitting energy levels for TiO 2 and spiro-omeTAD (shown in Fig. 10), which decreases the HOMO of TiO 2 and increases the LUMO of spiro-omeTAD.
How to obtain the external electric field in practice? As a preliminary exploration, we believe that this study is valuable. Indeed, the calculated results are aimed for the orthorhombic structure and not the tetragonal structure at room temperature, but the orthorhombic MAPbI 3 structure can be applied to solar panels in space with lower temperature. Due to cosmic microwave background with 3 K 29 , the perovskite solar cells with orthorhombic phase can accomplish as Power generation device candidate in space such as International Space Station, satellite, space shuttle, spacecraft, lunar rover vehicle, etc. Moreover, this may provide a new idea to control the properties of the tetragonal structure at room temperature.
The main reason for the orthorhombic to tetragonal transition in CH 3 NH 3 PbI 3 is temperature, and the external electric field causes the Stark effect splitting energy level of C, N, H, Pb and I, and the two structures with the same element and the similar chemical bond characters. Thus, we inferred that the Stark effect for CH 3 NH 3 PbI 3 with a tetragonal structure may be observed at room temperature.   Table 2. Related data obtained using eq (1). Here, E is the external electric field, E g is the band gap based on the band structure, λ is the wavelength, ν is the frequency calculated by the band gap, ν 100 is the frequency along the [100] direction based on the Fig. 8, and E g ′ is the band gap based on the optical properties. The device for realizing the external electric field is shown in Fig. 11, which is similar to the ref. reported by Li et al. 30 . It noted that the device in this study only provide the external electric field, do not provide electrons. The large DC voltage near 100 V can be obtained by DC boost circuit, which can resolve the electric field strength. The positive electrode is connected with the transparent ITO for visible light, and the negative electrode is connected with Au; the directions of the two electrodes can control the direction of the electric field, including the [111] direction. Moreover, the distance between the positive and negative electrodes, as well as the insulation thickness and dielectric constant, influenced the electric field strength.
It noted that the external electric field of about 0.06 eV/Å/e is large, which is equal to 0.6 V/nm, but this external electric field can be carried out in practice. For example, Hsu et al. used an external magnetic field to adjust the relative energy levels between a skyrmion and a ferromagnet globally, and obtained an electric field of 1 V/nm, which corresponds to a magnetic field of about 40 mT for their system 31 . Qin et al. applied an external electric field to drive the ultra-low thermal conductivity of silicene. Using an electric field (E z = 5 V/nm), the lattice thermal conductivity of silicene can be reduce a record low value of 0.091 Wm −1 K −1 , which is comparable to that of the best thermal insulation materials 32 . The main problem caused by the huge electric field may be the device breakdown, which is attributed to the self-sustainable discharge for uniform electric field. The more uniform the electric field is, the higher the self-sustainable discharge voltage is. So the transparent ITO as the positive electrode and the Au film as the negative electrode should possess the rule shapes and flat surfaces, which would decrease the degree of dielectric polarization, and reduce the possibility of the tip discharge.

Conclusion
In conclusion, we have used first-principles calculations to calculate the geometries, band structure, electronic properties, and optical absorption properties of perovskite CH 3 NH 3 PbI 3 under an external electric field aligned along the [111] direction. The external electric field increases the lattice parameters and the cell volume, stretching the c-axis and influencing the degree of lattice distortion. The external electric field controls the band gap  Methods CH 3 NH 3 PbI 3 perovskite undergoes two phase transitions, one at 160 K (orthorhombic to tetragonal) and the other at 330 K (tetragonal to cubic). The orthorhombic CH 3 NH 3 PbI 3 structure could be closer to that at the 0 K, so we chose this as the research object given the DFT calculations are performed at 0 K. Based on the structure reported by Menéndez-Proupin et al. 25 , CH 3 NH 3 PbI 3 perovskite has an orthorhombic crystal structure in the space group Pnma (no. 62), with lattice parameters of a = 8.8273 Å, b = 12.6793 Å, and c = 8.5099 Å, as shown in Table 3 and Fig. 12. CH 3 NH 3 PbI 3 has a typical AMX 3 perovskite structure with the unit cell consisting of a central lead atom octahedrally coordinated to six iodide atoms. The PbI 6 octahedron is located inside a cube with each iodide at the centre of a cubic face, and the CH 3 NH 3 cations are positioned at the corners of the cube. To study the effect of the electric field on the physical properties of CH 3 NH 3 PbI 3 , the external electric field along different directions (x, y, and z) was investigated.
We calculated the physical properties including the energy band, the density of states (DOS), and optical absorption in the orthorhombic perovskite CH 3 NH 3 PbI 3 exposed to different external electric fields using the Cambridge Serial Total Energy Package (CASTEP) 33,34 program. The generalized gradient approximation (GGA) functional developed by Perdew, Burke, andErnzerhof (1996, 2008, PBE, and PBEsol) 23,24 was used. These exchange-correlation functionals employed were same as those in the study of the self-regulation mechanism for charged point defects in hybrid halide perovskites reported by Walsh et al. 35 . The spin-orbit coupling (SOC) effect is reported to have little influence on the geometric structures 36 . Ultra-soft pseudopotentials with a cutoff energy of 310 eV (based on test results) were used to describe the interactions between the valence electrons and the ionic core, and including relativistic effects for Pb and I atoms. A 3 × 2 × 3 Monkhorst-Pack k-point scheme was used to calculate the absorption spectra. We performed the convergence test, and found the results with 500 eV cutoff energy and 5 × 5 × 5 Monkhorst-Pack k-point scheme to be similar to those with the above parameters. The convergence tolerances for geometry optimization calculations were set to a maximum displacement of 5.0 × 10 −4 Å, maximum force of 0.01 eV/Å, maximum energy change of 5.0 × 10 −6 eV/atom, maximum stress of 0.02 GPa. Figure 11. Device for realizing an external electric field. Here, V CC is input voltage obtained by the storage battery charged by the solar battery, L is inductance, u e is rectangular square wave pulse signal which control the duty cycle and regulate magnification, IGBT is crystal oscillator, D is diode, R is resistance, and C is capacitance.