## 3 Results and Discussion
### Disordered Structural Properties  
Based on the obtained ground state lattice constants of CZTSe, \(2\times 1\times 1\) supercells of the conventional kesterite unit cell have been generated, each containing four formula units (32 atoms). Within the Cu-Zn planes in these \(2\times 1\times 1\) supercells of the ordered kesterite structure we now find four Cu and four Zn cations, occupying the (2c) and (2d) Wyckoff positions (Figure 1a), or equivalently the (4d) Wyckoff position in the disordered kesterite structure (Figure 1b), respectively. The _special quasirandom structures_ have been generated by the supercell program of [PERSON] et al. [37], reducing the original \(\binom{8}{4}=70\) different  
Figure 2: (**a**) Total energy curves for the kesterite crystal structure of Cu\({}_{2}\)ZnSnSe\({}_{4}\) calculated by means of the SCAN (black) and the hybrid HSE06 (green) exchange and correlation functionals. In addition, the total energy curve obtained by the newly devised approach to speed up hybrid functional calculations is shown (red). (**b**) Comparison of the obtained ground state volumes with respect to experiment [2]. possible cation arrangements within the Cu-Zn planes to just 10 symmetry inequivalent cation distributions, shown exemplary in Figure 3.  
The middle panel of Figure 3 shows the total energy curves for each of the 10 symmetry inequivalent cation arrangements, calculated by means of the newly devised method to speed up hybrid functional calculations as discussed in Section 2.2. The total energy curves are normalised to one formula unit (f.u. ), and the lowest total energy is set to zero energy, respectively. Additionally, the dashed green line in the middle graph shows the weighted average over the 10 different total energy curves, taking into account the degeneracy of the symmetry inequivalent cation distributions, i.e., how often they appear in the total number of 70 possible cation arrangements (also shown for each of the considered cation arrangements in square brackets), while the dashed red line refers to the total energy of the ground state crystal structure of the ordered kesterite, respectively. Three conclusions can be drawn immediately from Figure 3: (1) there is a large spread in total energies for the 10 symmetry inequivalent cation distributions, reaching 238 meV per f.u. between the lowest and highest energy solutions, thus making the weighted averaging over all total energy curves a necessity to adequately model the total energy curve for the disordered kesterite crystal structure; (2) the energy difference between the total energies of the ordered (dashed red line) and the disordered (dashed green line) kesterite crystal structure amounts to 109 meV, making the ordered kesterite crystal structure. Figure 3: \(2\times 1\times 1\) superellis depicting the 10 symmetry inequivalent cation distributions of Cu (red) and Zn (green) over the (2c) and (2d) Wyckoff positions as generated by the supercell program of [PERSON] _et al._[37]. The lowest supercell depicts the ordered kesterite crystal structure, and the total energy differences \(\Delta E\) are given in meV and per formula unit for CZTSe including the respective degeneracy of the supercell in square brackets. Utilising the newly devised method to speed-up hybrid functional calculations resulted in total energy curves, as shown in the middle figure, where the grey lines correspond to the 10 symmetry inequivalent cation distributions, the dashed green line is their weighted average, and the dashed red line shows the result from the ordered kesterite crystal structure for comparison. The crystal structure figures have been prepared using VESTA [21].