Crystal structure across the β to α phase transition in thermoelectric Cu2−xSe

The average structure of β-Cu2−xSe is reported based on analysis of multi-temperature single-crystal X-ray diffraction data, and structural changes, including a large negative thermal expansion, across the β to α phase transition are discussed. The structural model also describes well high-resolution synchrotron powder X-ray diffraction data.

In the following the steps in analyzing the collected SCXRD data are elaborated. The data collected at 295 K are chosen as an example since the data obtained at 100 K are slightly twinned complicating the procedure, as seen in Figure S4.   (Milat et al., 1987, Lu et al., 2015 TGulay ( ) constructs a trigonal unit cell (using the hexagonal unit cell setting) from the reduced cell giving the cell parameters: a = 4.129(2) Å, b = 4.1397(15) Å, c = 20.50(1) Å,  = 90.02(3)°,  = 90.20(4)°, = 120.07(4)° and V = 303.3(2) Å 3 . Ceciprocal space including this unit cell together with all peaks Int > 1000 are depicted below. Notice the same orientation as the figure above.
Again a significant number of peaks are located at different fractions along the reciprical unit cell axes, which can be used to construct the suggested trigonal cells from the literature using simple transformation matrices.
T Vučić(1981) Table S2 shows the results of 4 selected integrations using different unit cell parameters and space-group symmetries. The integrations have been selected due to their relatively low Rint and are in the following discussed in the context of possible structure solutions. The resulting structural models are not based on any model from the literature. The diffraction images were first integrated without a lattice extinction filter and outlier rejection. For the background evaluation the "smart background" was applied with frame range 1, evaluation range 15 and repeat frequency 15. During the absorption correction the outlier rejection was used and the Fridel pairs set to be equivalent. The applied absorption was the automated empirical absorption correction.

Figure S6
Image of the same crystal mounted on a loop using paratone oil (left), for the LT 100 K and 295 K experiments, and mounted on an amorphous glass rod in Epoxy, two component glue, (right) for the HT experiments.

S3.3. Structure solutions
Structure solutions giving acceptable refinement parameters are found for the two first entries in Table S2 (1, 2) including all main reflections. It should be noted that the two refinements integrate precicely the same reflections. The resulting structural arrangements are also identical, with the resulting R1 factor being the same. Since solution 1 have a higher symmetry (trigonal vs monoclinic) we have choosen to continue with this structure. For the structural model, the only constraint is on the total Cu occupancy, which is fixed to the stochiometry found by the elemental analysis. Solution 2 has a lower Rint, but this is likely just a result of the lower symmetry. Slices of reciprocal space for 1 are shown below (100 K) illustrating the number of peaks not indexed.  In Figure S6b the encircled reflections are not indexed using the unit cell from solution 1 and 2, but are included using solution 3 which have the same monoclinic unit cell parameters as 2 but no C-centered. The extra peaks in Figure S6a are not included in 3. As seen from Figure S8 below the reflections breaking the C-centering are very weak. A satisfactory structural model could not be obtained from integration 3.

Figure S9
Slices of reciprocal space ((0kl) using the monoclinic integration 2,3 or 4) for the diffraction data obtained at 100 K, 295 K, and 372 K using an exposure time of 140 s, 60 s and 70 s, respectively. The contrast is set to 1k in all three diffractograms. The 1x3x4 supercell (4) indexes all the shown reflections at 100 K, while all reflections at 372 K can be indexed using the small monoclinic cell 3. Inserts show the peak size of the 4 peaks marked using blue circles. The weak powder rings originate from the instrument and not the sample.
Using integration 4 (integrating the same reflections as the unit cell suggested by Gulay et al. (2011)), which is a 1x3x4 super cell of the unit cell in integration 2, integrates the main part of all weak superstructure reflections (by removing the C centering additional peaks can be included in the integration) A structural model is obtainable from the integration with a resulting R1 value of11.8 % at 295 K. From Figure S8 it can be seen that the superstructure reflections are dissapearing at 372 K. The intensity of the main peaks in the figure at 295 K and 372 K have comparable intensities. It should also be noted that the along the c-axis there seems to be some indications of diffuse scattering.
From Figure S6a, if we go back to the trigonal unit cell, it looks like a trigonal 3x3x1 supercell could index all the reflections in the figure. However, this is the result of the diffuse scattering oriented along the c-axis.
The super-structure peaks do not have maxima in the depicted layer.
In summary, the symmetry of the main reflections is trigonal, while it is likely that the symmetry of the complete structure including superstructure peaks are not trigonal. In addition there seems to be diffuse scattering along the c-axis. The diffuse scattering along the c-axis explains why there are so many unassigned peaks along this direction in the reciprocal space after peak indexing (from the peak hunting).

S3.4. Possible super-structure models
The Gulay model (Gulay et al., 2011): In 2011 Gulay et al. studied Cu2-xSe using SCXRD and PXRD. They collected and integrated SCXRD data using space group C2/c and unit cell parameters a = 7.1379(4) Å, b = 12.3823(7) Å, с = 27.3904(9) Å, β = 94.308º. The unit cell integrates the same reflections as integration 4 (a = 7.152(1) Å, b = 12.384(1) Å, c = 28.817 (7), β = 108.84(2)º) with the following transformation matrix relating the two unit cells: The only information Gulay et al. writes concerning their structural model from SCXRD is the following: "A model of the structure [LT-Cu2-xSe] was obtained from X-ray single crystal diffraction data (R1 ≈ 0.14) at room temperature. At the second step, X-ray powder diffraction data were used for the refinement" Since the authors were not satisfied with the high R1 value of 0.14, indicating that their structural model do not fit their diffraction data, they instead used the model on their PXRD data resulting in a R1 value of 0.0765.
It should be stated here that this procedure of using an incorrect model based on SCXRD to fit PXRD data is not considered acceptable. Obviously a correct structural model has to be able to describe the SCXRD data.
That being said it is informative to see how their proposed model fits our collected SCXRD data.
Unfortunately, they only publish their structural model after it has been refined against their powder diffraction data. The comparison is therefore based on this structure. The TGulay,4 matrix has been used in order to get the correct fractional coordinates for the reflection list from integration 4.

Table S3
The Gulay model taken from (Gulay et al., 2011) with fractional coordinates recalculated for the unit cell a = 7.152 (9)   were also refined. Furthermore, the list of reflections have been divided into main and super-structure reflections in order to see how the individual models fit the main and super-structure peaks, respectively.
From the tabulated data it is evident that the Gulay model does not fit our SCXRD data, neither the main reflections nor the super-structure peaks. Even when freely refining all atomic positions the overall R1 value is still 0.294. It should be noted that the refinements are all stable, but there is a lot of electron density  In summary, it is evident that the structural model proposed by Gulay et al. (2011) does not fit their own original SCXRD data (R1 ≈ 0.14) nor does itfit the SCXRD data presented here (R1 = 0.294).

S3.5. Ordered super-structure
If the suggested ordering in section 6 (main paper) is long-range order then a possible super-structure model can be constructed. The ordering is depicted in Figure 5b & d and the fractional coordinates are given below.

S3.6. Disordered super-structure
By introducing disorder into the structure the model nicely fits the main reflections (R1 = 0.062), while the model do still not fit the super-structure peaks well (R1 = 0.253), resulting in a total R1 of 0.119 when using all reflections. It should be noted that the refinements are quite stable with a nice observable to parameters ratio. The misfit is therefore not the result of unstable refinments or too many parameters.
Furthermore, the max/min residual electron density only amount to 2 electrons, indicating that there is no large electron density regions not accounted for ( Figure S11). However, the model does not fit the superstructure peak intensities well.
In order to progress and getting a better structural model we believe it is important to use longer exposures focusing on getting better I/σ ratios for all super-structure reflections. In practice this is only feasible using high brilliance synchrotron facilities. From the integration above, 1348 out of the 2226 unique reflections have I/σ < 2.
Better intensities for the super-structure reflections will hopefully allow for a determination of the correct space group symmetry of the super-structure. In the end it might be necessary to use superspace formalism in order to describe the complex super-structure.

S4. Supercell reflection intensities, with temperature.
In order to evaluate the intensity of the superstructure peaks we have chosen integration 4, for the 100 K, 295 K and 372 K data. The large 1x3x4 monoclinic supercell integrates a large number of weak superstructure peaks when the conditions k/3 ≠ n and l/4 ≠ n are fulfilled. In summary, the intensity of the superstructure peaks decrease with respect to the intensity of the main peaks (a factor 5), as we approach the phase tranition. Thus when approaching the phase transition the superstructure approaches the average structure.

Figure S15
Known structures of β-Cu2-xSe; Se and Cu are represented by green and blue spheres, respectively, and the structures have been oriented to show the similarities. a) S1 structure from Lu et al., (2015), with a stabilizing energy of -0.2921 eV compared to the cubic phase. b) S2 structure from Lu et al., (2015), with a stabilizing energy of -0.2920 eV compared to the cubic phase. c) S3 structure from Lu et al., (2015), with a stabilizing energy of -0.2990 eV compared to the cubic phase. d) The structure from Gulay et al., (2011). e) The structure from Nguyen et al., (2013). f) The β-Cu2-xSe structure determined from PXRD Rietveld refinements of data collected at 300 K. The partly colored portion of the Cu atoms indicates the occupancies for the different copper sites. The purple ellipse marks a structural "unit" that can easily be compared and found in all the shown structures.

S6. Synchrotron powder X-ray diffraction
The theoretical diffraction pattern calculated for the structural model obtained by Rietveld refinement is shown in Figure S11 together with the data and difference curves for the 300, 200 and 100 K data.