Static and dynamic components of Debye–Waller coefficients in the novel cubic polymorph of low-temperature disordered Cu2ZnSnS4

Disorder greatly affects transport properties, as observed for Cu2ZnSnS4. A novel cubic polymorph of Cu2ZnSnS4 is characterized, and cation disorder is observed in a static contribution to Debye–Waller coefficients.

Static and dynamic components of Debye-Waller coefficients in the novel cubic polymorph of low-temperature disordered Cu 2 ZnSnS 4 Supplementary Material SN 1: Additional TEM and SEM images, and elemental analysis Figure S2. Transmission electron microscopy images and site-specific high-magnification energy dispersive X-ray spectroscopy for a CZTS sample from mechanical alloying treated at 833K. Although the elemental analysis is not calibrated, measurements on areas 1 and 2, likely representing individual domains, show the presence of all the CZTS elements. Areas 3 and 4 clearly show the formation of Sn oxide nanoparticles, as also spotted by XRD. This can explain the Sn depletion observable in the stoichiometry of Areas 1 and 2. Figure S1. Transmission electron microscopy images for as-milled CZTS samples from mechanical alloying (a), and treated at 573K (b,c).

Unit cell transformation
In the crystal structure of cubic CZTS, a stacking of layers of corner sharing (Cu/Sn/Zn)S4/2 tetrahedra occurs along the [111] direction. Using DIFFaX-like recursive supercell approaches requires a unit cell transformation ( Figure S5) into an either pseudo-orthorhombic or trigonal cell, where the layers are stacked along the crystallographic c-axis. The lattice parameters, as well as the atomic coordinates of the transformed unit cell are given in Table S2.

Faulting scenarios
The crystal structure of cubic CZTS consists of a cubic close packing of S anions and occupationally disordered Cu, Zn and Sn cations situated in the tetrahedral interstitial voids. The transition from a layer to the subsequent one can be described by using the stacking vector S1 ⃗⃗⃗⃗ . Within the faultless structure the stacking occurs in a homogenous fashion, i.e. the stacking vector does not change (see Error! Reference source not found. in the main text and Figure S6). Twinning describes a prominent faulting scenario for crystals, where differently oriented crystallites are intergrown. For the CZTS cubic structure, this means that the stacking order of the staggered sulfide layers is inverted and that the metal cations occupy a neighboring, tetrahedrally coordinated interstitial void. A single twin consists of two homogenously stacked sections with different stacking vectors (S1 ⃗⃗⃗⃗ and S2 ⃗⃗⃗⃗ , Figure S6). In a polycrystalline sample twinning can occur multiple times with a random distribution of twin boundaries. An alternating arrangement of differently oriented stacking vectors, i.e. of S1 ⃗⃗⃗⃗ and S2 ⃗⃗⃗⃗ leads to a hexagonal close packing of the anion substructure. This can also occur as an isolated fault, which is comparable to a dislocation in the layer stacking. For the investigations of the microstructure of cubic CZTS we considered both twin and hexagonal faults. As the metal cations occupy different tetrahedral interstitial voids after an inversion of the stacking order of the anions, two layer types, consisting of one anion and one cation, were defined. The basic faultless cubic stacking was defined by combining layer type A with stacking vector S1 ⃗⃗⃗⃗ . A change from layer type A to type B also occurs via the stacking vector S1 ⃗⃗⃗⃗ and is connected to a fault probability (from 0 to 1), px for twin faults and py for hexagonal faults. The probability for a continuation of the faultless stacking order equals to one minus the fault probabilities. In the twin faulting scenario, a stacking of layer type B via stacking vector S2 ⃗⃗⃗⃗ is considered as the new basic stacking order and switching back to layer type A is considered as an additional fault. For a hexagonal fault, the stacking switches immediately (probability = 1) back from layer type B to layer type A. We considered a combination of twin and hexagonal faults as well. A graphical representation of the faulting models is given in Figure S7 and the stacking vectors, layer setups and the transition probability matrixes can be found in Tables S4 and S5.

SN 6: Compliance tensors and anisotropy in CZTS
In order to measure the anisotropy in CZTS, we performed DFT calculations using the finite difference method. This allows us to obtain the compliance tensor for the ordered and disordered polymorphs, which are given respectively as, where all values are in GPa.
It should be noted that to minimize the forces acting on individual ions, the DFT reduces the symmetries of the supercell to triclinic. This leads to small non-zero values of cij for i,j > 3 (i≠j). In addition, given the metastable nature of the disordered polymorph, several negative terms are also introduced. Nevertheless, this does not significantly affect the overall qualitative nature of the anisotropy in the two polymorphs, or their comparison. For the ordered tetragonal polymorph, the compliance matrix is in good agreement with that obtained by He and Shen (He & Shen, 2011). In order to quantify the degree of anisotropy, we use the log-Euclidean anisotropy parameter A L (Kube, 2016), which is valid for all crystalline symmetries. It is an absolute measure of anisotropy, with perfect isotropy yielding a value of zero. This measure of anisotropy is given as, where C is the compliance tensor, and are the bulk and shear moduli respectively, while the superscripts V and R refer to the Voigt and Reuss averages.
For the ordered tetragonal polymorph, this parameter has a value of 0.77 while for the disordered cubic this comes out to 1.12, indicating that the latter polymorph is more anisotropic. This of course is in agreement with the fact that full cation disorder leads to significant inhomogeneous bonding in cubic CZTS. In comparison to SrVO3, which is the most anisotropic cubic crystal shown in Ref #, with a value of 5.3, CZTS presents a relatively limited anisotropy.
In order to further understand the nature of the anisotropy, we have calculated the Young's modulus projected along the [hhh] and [h00] family of directions. The reciprocal of Young's Modulus in the direction of the unit vector li for the general triclinic system is given by (Nye, 2006), SN 7: Low-temperature XRD data Figure S9. XRD in temperature for CZTS samples from mechanical alloying treated at 573K (a) and 833K (b).
SN 8: Further data about Rietveld refinements of XRD patterns in temperature SN 9: Reliability of Debye-Waller coefficients Figure S9 shows a comparison of reliability between Biso values for the cation and the anion. The trend of the refinement Rwp for an imposed variation in Biso, clearly shows that the fit is less sensitive towards a variation in the anion Biso. The minima are instead more clearly identified for the cation Biso. Furthermore, the estimated standard deviations (known to be an underestimation of the real value error) are found between 2 to 3 times larger for the anion's value than the cation's. These features, coupled with the known lower accuracy of XRD in determining the atomic displacement parameter for lighter elements, point to a muchreduced credibility of the anion values. This can explain the stronger fluctuation of experimental values which is observed between different measurements and datasets. In addition, we do not exclude possible sample deviations between the high-and the low-temperature datasets, as the specimens were different. Figure S10. Temperature evolution of lattice parameters (a,b), Debye-Waller coefficients (c,d) and faulting probability (from 0 to 1) (e) from the Rietveld refinement of XRD data for samples treated at 573K (a,c,e) and 833K (b,d). In (a,b,c,d) the effect of imposing a linear relationship with temperature or leaving the parameters free is compared.