Establishing the limits of efficiency of perovskite solar cells from first principles modeling

The recent surge in research on metal-halide-perovskite solar cells has led to a seven-fold increase of efficiency, from ~3% in early devices to over 22% in research prototypes. Oft-cited reasons for this increase are: (i) a carrier diffusion length reaching hundreds of microns; (ii) a low exciton binding energy; and (iii) a high optical absorption coefficient. These hybrid organic-inorganic materials span a large chemical space with the perovskite structure. Here, using first-principles calculations and thermodynamic modelling, we establish that, given the range of band-gaps of the metal-halide-perovskites, the theoretical maximum efficiency limit is in the range of ~25–27%. Our conclusions are based on the effect of level alignment between the perovskite absorber layer and carrier-transporting materials on the performance of the solar cell as a whole. Our results provide a useful framework for experimental searches toward more efficient devices.

A hand waving measure of stability against distortions of the octahedral cage can be obtained by allowing full relaxation of all parameters of the unit cell. Table 2 displays the relative bond distortion in the octahedron. The standard deviation of the B---X---bond lengths should be low if the octahedron is stable. Our calculations show that GUSnBr3 is most likely unstable as a result of the unfavorable size ratio of the GU molecule to the rather small Sn and Br ions. GUSnBr3 is omitted from subsequent calculations due to the low likelihood of producing the compound in the pseudo---cubic structure.
Visualizations and manipulations of the crystal structure is done with the software packages VESTA and cif2cell 9,10 .

Band structure and effective carrier masses
Valence and conduction band masses are calculated using a fixed charge density, with a dense k---point mesh around the R---point in the Brillouin zone. The band masses are converged at a sampling density of approximately 2×10 % k---points per Å ---3 . The masses reported are well in line with recent experimental work 11 slightly lower than what is reported in other calculations 12---14 . In modelling the efficiency, only its temperature dependence is affected by the effective masses.
The density of states for all compounds is shown in Fig. 1

Supercells for level matching
In order to determine the relative shifts in valence and conduction band between the perovskites we use a supercell approach. Five calculations for each pair of compounds has to be considered. First, the super lattice, setup according to Finally, when the strained cell AP to VBM is known, we can calculate the corresponding parameters for the relaxed cell. We now have access to the full relative difference, the natural band alignment, of the two constituents of the superlattice using the formula: where 4E7 is the external bias, Δ E7V is the potential drop in the junction to the electron and hole transporting materials, E * , E is the respective carrier mass and density for electrons and holes.
Conservation of current densities assures that 4E7 = @G − 34B , that is, the generated current gained from absorption of the photon flux, and loss though carrier recombination. Ideally the only recombinative losses in the system are from black body radiation, although in Silicon impurity assisted (Shockley---Reid---Hall (SRH)) recombination is often dominant. Due to the fact that the perovskites have proven remarkably resilient to material defects, in this work we assume that SRH recombination rates are small in comparison to band---to---band recombination. Previous work by Miyano and co---workers assumed a two diode model where SRH recombination occurs at the interface only. 19 The radiative recombination rate is calculated from Since all the compounds under consideration have a substantial band---gap, we approximate this expression with Inserting equation (1) in this expression, we have the relation between the radiative recombination rate and the external bias. The expression for the JV characteristics of the PIN junction is The number of absorbed photons @G is calculated assuming all photons above the band---gap are absorbed, that is, the cell is sufficiently thick to eliminate transmission.  Fig. 3. Inspiration for the python implementation was acquired from S. J. Byrnes pedagogic IPython notebook [20] .