Enhanced optical absorption via cation doping hybrid lead iodine perovskites

The suitable band structure is vital for perovskite solar cells, which greatly affect the high photoelectric conversion efficiency. Cation substitution is an effective approach to tune the electric structure, carrier concentration, and optical absorption of hybrid lead iodine perovskites. In this work, the electronic structures and optical properties of cation (Bi, Sn, and TI) doped tetragonal formamidinium lead iodine CH(NH2)2PbI3 (FAPbI3) are studied by first-principles calculations. For comparison, the cation-doped tetragonal methylammonium lead iodine CH3NH3PbI3 (MAPbI3) are also considered. The calculated formation energies reveal that the Sn atom is easier to dope in the tetragonal MAPbI3/FAPbI3 structure due to the small formation energy of about 0.3 eV. Besides, the band gap of Sn-doped MAPbI3/FAPbI3 is 1.30/1.40 eV, which is considerably smaller than the un-doped tetragonal MAPbI3/FAPbI3. More importantly, compare with the un-doped tetragonal MAPbI3/FAPbI3, the Sn-doped MAPbI3 and FAPbI3 have the larger optical absorption coefficient and theoretical maximum efficiency, especially for Sn-doped FAPbI3. The lower formation energy, suitable band gap and outstanding optical absorption of the Sn-doped FAPbI3 make it promising candidates for high-efficient perovskite cells.

Over the last several years, hybrid organic-inorganic perovskite solar cells have become one of the most attractive photovoltaic technologies, with easy solution fabrication and high conversion efficiencies [1][2][3][4][5][6][7][8] . The first perovskite based solar cells, made seven years ago by Japanese researchers, turned just 3.8% of the energy in sunlight into electricity 9 . After that, the efficiency of perovskite solar cells has been updated rapidly as a result of new strategies adopted in their fabrication process [10][11][12][13][14][15][16][17][18] , including device structure, interfacial engineering, chemical compositional tuning, and crystallization kinetics control. The power conversion efficiency of perovskite solar cells can greater than 20% 18,19 , which is comparable to the commercial silicon (20%), CIGS (19.6%), GaAs (18.4%) and CdTe (19.6%) solar cells 20 . More recently, a power conversion efficiency up to 22.1% under the operational condition is achieved 21 . The power conversion efficiency of perovskite solar cells is climbing faster than that of any solar technology before them.
The outstanding light absorption is one of the indispensable conditions for the high efficiency solar cells. While, the band gap plays a vital role of light absorption. If the band gap is too small, the device will be able to collect extra current but the open-circuit voltage will be too small. However, if the band gap is too wide (>2 eV), only a small fraction of solar energy can be absorbed. Thus, an absorbing layer with a band gap of approximately 1.4-1.6 eV is preferred for solar cells developed from a single junction 22 . Perovskite materials are built by the inorganic elements lead and iodine, together with simple organic compounds. Most of previous works are mainly focused on the methylammonium lead iodide (MAPbI 3 ) perovskites, with a band gap of ∼1.55 eV [23][24][25][26][27][28][29] . Compared with un-doped MAPbI 3 perovskite, Sn-doped MAPbI 3 perovskite have a small band gap, which can further enhance the photovoltaic performance of perovskite solar cells in the near-infrared spectrum 30,31 . Besides, Sn-doped MAPbI 3 perovskite allowed tunable band gap of the perovskite absorber by varying the Sn:Pb ratio 32,33 .
Once replacing organic compound methylammonium (MA) with formamidinium (FA), a slightly larger organic molecule, the absorption spectrum of perovskite is mostly concentrated in the visible and near-infrared regime [34][35][36] . Especially for the tetragonal FAPbI 3 perovskite with a band gap of 1.43 eV, which is therefore potentially superior than the trigonal FAPbI 3 as the light harvester 35 . Besides, the FA induced structural variability improved charge transport and red-shifted absorption in tetragonal FAPbI 3 structures 36 . More importantly, the highest confirmed record power conversion efficiency of PSCs is based FAPbI 3 perovskite 18 . We note that the FASnI 3 has a band gap of 1.41 eV which allows light harvesting from the near-infrared region 37 . Thus, we very curious to know the electronic structures and optical absorption properties of Sn and other cation-doped tetragonal FAPbI 3 perovskite.
In this work, first-principles calculations were carried to systematically examine the geometry, electronic structure, and optical properties of the cation (Bi, Sn, and TI) doped tetragonal MAPbI 3 /FAPbI 3 perovskites. The formation energies and detailed defect-I bond lengths of cation-doped MAPbI 3 /FAPbI 3 are showed in the Table 1. The calculated results show that the Sn-doped defect is the common impurity in the tetragonal MAPbI 3 / FAPbI 3 perovskites due to the lowest formation energy of about 0.3 eV. While, relatively higher formation energy means that both the donor defect Bi and the acceptor defect TI are difficult to dope in MAPbI 3 /FAPbI 3 perovskites. The calculated band gap of Sn-doped MAPbI 3 /FAPbI 3 perovskite is 1.30/1.40 eV, respectively. The band gap of Sn-doped MAPbI 3 /FAPbI 3 perovskite is smaller than that of un-doped perovskites. More importantly, the Sn-doped MAPbI 3 and FAPbI 3 perovskites have the higher specific absorption in the visible light region, especially for the Sn-doped tetragonal FAPbI 3 perovskite. Our electronic structures and optical properties calculations indicate that the Sn-doped tetragonal FAPbI 3 is a promising candidate for high-efficient perovskite cell.

Results
Before the optimization of the cation-doped perovskite structure, the lattice constants of tetragonal MAPbI 3 and FAPbI 3 supercells are fully relaxed. In the tetragonal MAPbI 3 supercell, the relaxed lattice constants a is 8.72 Å and c is 12.92 Å, which is in good agreement with the experimental results 32 . In addition, the relaxed lattice constants a is 9.20 Å and c is 12.54 Å in the tetragonal FAPbI 3 supercell. Then, we fix the lattice constants in the structural optimization of the cation-doped MAPbI 3 /FAPbI 3 supercells. Considered the ion radius and the number of outside electrons, three types of atoms (Bi, Sn, and TI) were chosen as the typical cation-doped in the MAPbI 3 and FAPbI 3 . The Bi and TI represent the donor impurity and acceptor impurity, respectively. The outsider electron number of Pb equal to the Sn atom, which is neither an acceptor impurity, nor a donor impurity. The relaxed structures of Sn doped MAPbI 3 and FAPbI 3 supercells are showed in Fig. 1(a) and (b). For the Sn-I octahedral structure in the Sn doped MAPbI 3 , the average horizontal and vertical Sn-I bond length is 3.13 Å and 3.21 Å, respectively. The Sn-I octahedral structure in the Sn doped MAPbI 3 is a tensile octahedron, which vertical Sn-I bond length larger than the horizontal Sn-I bond length. However, the vertical and horizontal Sn-I bond length  Where the E f , L-H defect-I , and L-V defect-I are the formation energies, the average defect-I bond lengths in the horizontal surface of defect-I octahedral structure and the average defect-I bond lengths in the vertical direction of defect-I octahedral structure, respectively. To get a deeper understanding the electronic properties of the cation-doped MAPbI 3 /FAPbI 3 , the total density of states (DOS) and partial density of states (PDOS) of Pb, I and cation defects are plotted in the Fig. 3. The partial DOS of Bi-doped MAPbI 3 /FAPbI 3 shown that the electronic states near the Fermi level are mainly contributed by Bi defects, as shown in the Fig. 3(a) and (d). For the Sn-doped MAPbI 3 /FAPbI 3 , the valence band maximum (VBM) is mainly contributed by I atom, while the conduction band minimum (CBM) is mainly contributed by Sn and Pb atom. Besides, most PDOS of Sn defect are overlapped with the PDOS of single Pb atom becuase Sn and Pb have the same outer electron configuration. Compared with the PDOS of Pb atom, more sattes of TI in the TI-doped MAPbI 3 /FAPbI 3 distributed in the high energy region, which further confirms that the TI is an acceptor defect in the doped MAPbI 3 /FAPbI 3 systems.
To evaluate the optical absorption of halide perovskites, the optical absorption efficients of the Sn and TI doped MAPbI 3 /FAPbI 3 perovskites are calculated and compared with the un-doped MAPbI 3 /FAPbI 3 perovskites, as shown in Fig. 4. For the Sn-doped MAPbI 3 /FAPbI 3 , the optical absorption peak is lower than that of undoped MAPbI 3 /FAPbI 3 . However, the Sn-doped MAPbI 3 has better light absorption in the visible regions (380-780 nm), which is consistent with recent theoretical and experimental results. In contrast to the un-doped MAPbI 3 /FAPbI 3 , the optical absorption peak of Sn-doped MAPbI 3 /FAPbI 3 exhibits a red-shift. But the optical absorption spectrum of TI-doped MAPbI 3 /FAPbI 3 is lower than that of un-doped structures in most of the visible light region. At the strongest emission ares of sunlight (450-500 nm), the absorption efficient of Sn-doped FAPbI 3 is about in 1.5 × 10 6 cm −1 , which is 1.5 times larger than that of Sn-doped MAPbI 3 . Considering the range of visible light accounts for the major usable portion of the full solar spectrum, the visible light absorption is critical to achieve high efficiency cells. Therefore, it is very essential to know whether the Sn-doped FAPbI 3 can enhance the photoelectric conversion efficiency.
In general, the effect of the optical absorption coefficient is not considered in the well-known Shockley-Queisser limit 39 . The theoretical maximum efficiency depends on the thickness of the absorber layer [40][41][42] . Yin et al. 26,43 calculated the thickness-dependent maximum solar cell parameters of CH 3 NH 3 PbI 3 based on Fermi Golden rule. According to the Fermi Golden rule, the optical absorption of a photonic energy ħω is directly cor- , Where < > Ĥ v c is the transition matrix from states in the valence band (VB) to states in the conduction band (CB) and the integration is over the whole reciprocal space. For a real solar cell, the theoretical maximum efficiency depends on the thickness of the absorber layer 43 . After taking the absorption efficient and absorber layer thickness into consideration, we have calculated the maximum efficiencies of some common light absorbers as a function of the thickness of the absorber layers, as shown in Fig. 5. With a 5 µm absorber, the maximum efficiency of Sn-doped MAPbI 3 , TI-doped MAPbI 3 , un-doped MAPbI 3 , Sn-doped FAPbI 3 , TI-doped FAPbI 3 , and un-doped FAPbI 3 based cells is 32.4%, 25.0%, 31.3%, 32.7%, 29.3%, and 31.9%, respectively. Obviously, the Sn-doped MAPbI 3 /FAPbI 3 perovskites exhibit much higher conversion efficiencies than un-doped MAPbI 3 /FAPbI 3 and TI-doped MAPbI 3 /FAPbI 3 for any given thickness. More importantly, the Sn-doped MAPbI 3 /FAPbI 3 perovskites are capable of achieving high efficiencies with very thin absorber layers. For example, with a 0.5 µm absorber, Sn-doped MAPbI 3 and Sn-doped FAPbI 3 based cells can have a maximum efficiency up to 23.2% and 21.9%, respectively. Considering the strong capacity of light absorption and high maximum efficiency, the Sn-doped tetragonal FAPbI 3 should be a more suitable candidate for the high efficiency perovskite solar cell material.

Discussions
To know the difficulty of Bi, Sn, and TI doping in the MAPbI 3 and FAPbI 3 , the formation energy, E f , is calculated. The formation energy of the cation-doped MAPbI 3 /FAPbI 3 is defined as follows,   In this work, the electronic structures and optical properties of typical cation (Bi, Sn, and TI) doped MAPbI 3 / FAPbI 3 are studied by density functional theory. The calculation results show that both the donor defect Bi and the acceptor defect TI have the relatively high formation energies. While, the Sn defect is easy to dope in the tetragonal MAPbI 3 /FAPbI 3 structure due to the small formation energy of 0.3 eV. The calculated band gap of Sn-doped MAPbI 3 and FAPbI 3 is 1.30 eV and 1.40 eV, respectively. The optical absorption efficients of Sn-doped MAPbI 3 /FAPbI 3 are higher than that of un-doped MAPbI 3 /FAPbI 3 within the visible light range. More importantly, the Sn-doped MAPbI 3 /FAPbI 3 have relatively high theoretical maximum efficiency, especially for the Sn-doped FAPbI 3 . The lower formation energy, suitable band gap and outstanding optical absorption of the Sn-doped FAPbI 3 , enable it has great potential applications for the high-efficient perovskite cells.

Method
The first-principles structure, energy and optical absorption calculations were performed by the Vienna Ab Initio Simulation Package (VASP) 44,45 . Projector augmented-wave (PAW) pseudopotentials 46 were used to account electron-ion interactions. The generalized gradient approximation (GGA) with the PBE functional 47 was used to treat the exchange-correlation interaction between electrons. In order to get the appropriate doping concentration, 2 × 1 × 1 MAPbI 3 and FAPbI 3 supercells are used in our calculation. The energy cutoff was set to 500 eV and a 5 × 7 × 7 Monkhorst-Pack scheme was used to sample Brillouin zone 48 . The full geometry optimizations are carried out with the convergence thresholds of 10 −4 eV and 1 × 10 −2 eV/Å for total energy and ionic force, respectively. It is well-known that vdW interactions are crucial in the determination of the equilibrium configurations in the hybrid structure. Thus, the DFT-D3 approach was used to take the effect of the vdW interaction 49 .
It is well known that the PBE functional always underestimated the band gap of semiconductors. Besides, the spin-orbit coupling (SOC) also results in much reduced band gaps in hybrid lead iodine perovskite structure. In the previous DFT calculation, both the hybrid HSE06 functional and spin-orbit coupling effects are considered to calculated the electronic properties of hybrid lead iodine perovskite structure. Their calculated results show that the band gap of cubic MAPbI 3 is 1.60 eV with PBE functional, while the band gap of PBE + SOC and PBE + HSE + SOC is 0.49 eV and 1.53 eV 38 . It is noted that the band gaps obtained by PBE without including SOC is quite close experimental value of 1.55 eV 9 . Thus, the PBE functional could give the reasonable band gaps for hybrid lead iodine perovskites. In addition, the calculated optical of cation-doped MAPbI 3 /FAPbI 3 with PBE functional also should show the right trend.