Relativistic Solar Cells

Hybrid AMX3 perovskites (A=Cs, CH3NH3; M=Sn, Pb; X=halide) have revolutionized the scenario of emerging photovoltaic technologies. Introduced in 2009 by Kojima et al., a rapid evolution very recently led to 15% efficient solar cells. CH3NH3PbI3 has so far dominated the field, while the similar CH3NH3SnI3 has not been explored for photovoltaic applications, despite the reduced band-gap. Replacement of Pb by the more environment-friendly Sn would facilitate the large uptake of perovskite-based photovoltaics. Despite the extremely fast progress, the materials electronic properties which are key to the photovoltaic performance are relatively little understood. Here we develop an effective GW method incorporating spin-orbit coupling which allows us to accurately model the electronic, optical and transport properties of CH3NH3SnI3 and CH3NH3PbI3, opening the way to new materials design. The different CH3NH3SnI3 and CH3NH3PbI3 properties are discussed in light of their exploitation for solar cells, and found to be entirely due to relativistic effects.

Compared to MAPbI 3 , the analogous MASnI 3 perovskite has been much less explored. 8,9 The two compounds show a similar tetragonal structure 9 (although in different temperature ranges) but different optical properties, with MASnI 3 (MAPbI 3 ) having an absorption onset at 1.2 (1. 6) eV. 9,10 Experimental data also indicate that CsSnI 3 and MASnI 3 are excellent hole transporters, 6,9 while MAPb(I 1-x Cl x ) 3 and MAPbI 3 can sustain high rates of electron and hole transport, respectively. 3,7 Understanding the origin of the different electronic properties of AMX 3

materials, with
M=Sn and Pb, could represent a fundamental step towards the large-scale uptake of perovskitesbased photovoltaics. In this context, a first principles computational approach capable of reliably calculating the materials band-gap and electronic/optical properties, thus trustfully allowing to design new materials and to interpret their properties, is fundamentally required. While standard Density Functional Theory (DFT) provides reliable structures and stabilities of perovskites, [11][12][13] it considerably underestimates the band-gap of these materials and in general of semiconductors. DFT with asymptotically correct functionals partly overcomes this shortcoming. 13 Many body perturbation theory, within the GW approach, 14,15 although more computationally demanding, constitutes a solid framework to improve upon DFT. 12,16 Contrary to expectations, DFT-calculated band-gaps of MAPbI 3 were in surprisingly good agreement, within 0.1 eV, with experimental values. 17,18 For the supposedly similar ASnX 3 perovskites, DFT provided a 1 eV band-gap underestimate. 8,11,12,19 Such an unbalanced description of Sn- and Pb-based materials hampers any predictive materials design/screening or comparative interpretation of their properties.
The large calculated band-gap difference between ASnX 3 and APbX 3 perovskites might be due to relativistic effects, particularly strong in Pb. 20,21 Relativistic effects are usually approximated by scalar relativistic (SR) and, to higher order, by spin-orbit coupling (SOC) contributions. A recent DFT investigation has confirmed a relevant SOC in MAPbX 3 , leading to a strong, and opposite to the estimated GW correction, band-gap reduction. 22 This analysis poses the quest for a reliable and efficient theoretical framework for the simulation of ASnX 3 and APbX 3 perovskites and possibly of mixed Sn/Pb compounds. The method of choice is ideally a GW approach incorporating SOC. 23 A very effective GW implementation is also required, which was devised by some of us. 24 Here we develop a novel approach to introduce SOC effects into our efficient GW scheme. The resulting SOC-GW method is computationally affordable and it accurately reproduces the band-gap and electronic/optical properties of MASnI 3 and MAPbI 3 . To make a direct connection between our calculations and solar cell operation, in Figure  2 we report the maximum J sc which can be extracted from a solar cell employing a material of varying band gap. The agreement between our SOC-GW calculated band-gaps and the experimental ones allows us to estimate the maximum J sc within 10%. As an example, for MAPbI 3 we calculate a maximum J sc of 25 mA/cm 2 against a 28 mA/cm 2 value derived from the experimental band-gap.
It is also worth noticing the potential of the MASnI 3 material to deliver extremely high J sc values due to its reduced band gap. This characteristic, along with its good transport properties, make this material highly promising to replace MAPbI 3 , although some sensitivity of the material to the preparation conditions have been reported. 9 Top J sc values measured for solar cells based on MAPbI 3 stand at 21 mA/cm 2 . 2,5 The reason for the non-optimal photocurrent generation can be traced back to the reduced light harvesting efficiency measured in the 600-800 nm range ( 2.0 1.5 eV), 2 Figure 3. Based on our SOC-GW calculated electronic structure, we thus simulated the optical absorption spectrum of MAPbI 3 , albeit neglecting electron-hole interactions, Supplementary Information. The employed procedure was shown to represent a reasonable approximation to the optical spectra of small bandgap semiconductors. 26 The results are reported in Figure 3, along with experimental data for MAPbI 3 . The calculated data satisfactorily matches the experimental UV-vis spectrum: the bandgap absorption, the rise of the spectrum at higher energy and the feature at ~2.6 eV are nicely reproduced, despite the approximate spectral calculation. Compared to MAPbI 3 , the absorption spectrum of MASnI 3 shows a red-shift (in line with the reduced band-gap) and increased intensity, The VB structure of the investigated systems is relatively similar, although MASnI 3 shows a widening and structuring of the VB compared to MAPbI 3 due to states found within 1 eV below the VB maximum. The MAPbI 3 CB has a tail at lower energy compared to MASnI 3. Notably, in the absence of SOC the CB of MAPbI 3 has essentially the same structure as that of MASnI 3 , Supplementary Information. The reduced spectral intensity calculated for MAPbI 3 appears thus to be due to the comparatively lower DOS close to the CB bottom, which is due to SOC. We can also compare the relative VB/CB position with available experimental data for MAPbI 3  MAPb(I 1-x Cl x ) 3 compounds were found to efficiently transport both holes and electrons. 3,7 Our calculations also suggest MASnI 3 to be a potentially good electron transporter, in line with recent mobility results, 9 although to our knowledge this material has never been employed in solar cells.
We can also compare the calculated reduced masses =m e m h /(m e +m h ) with experimental data for

Method.
We have extended the relativistic DFT scheme of Ref. 28 , in which the spin-orbit coupling is included by 2-dimensional spinors and modeled by pseudopotentials, to our GW approach. Wavefunctions and charge densities are developed on a plane-waves basis sets. The two dimensional spinor exchange operator is expressed as: (1) where the index and run over the two spinor components of the occupied relativistic KS states . For evaluating the self-energy we have considered the suggestion of Ref. 23 of approximating the screened relativistic coulomb interaction with that obtained from a scalar relativistic calculation : For calculating the relativistic correlation part of the self-energy we can calculate the DFT considering explicitly only the lowest relativistic states: where for simplicity in the scalar relativistic calculation we have considered doubly occupied states.
In this way we still avoid sums over unoccupied KS states which would be particularly cumbersome when dealing with large model structures. The PBE exchange-correlation functional 29