Evidences of inner Se ordering in topological insulator PbBi2Te4-PbBi2Se4-PbSb2Se4 solid solutions.

In topological insulators (TIs), carriers originating from non-stoichiometric defects hamper bulk insulation. In (Bi,Sb)2(Te,Se)3 TIs (BSTS TIs), however, Se atoms strongly prefer specific atomic sites in the crystal structure (Se ordering), and this ordering structure suppresses the formation of point defects and contributes to bulk insulation. It has accelerated the understanding of TIs' surface electron properties and device application. In this study, we select Pb(Bi,Sb)2(Te,Se)4 (Pb-BSTS) TIs, which are reported to have larger bandgap compared to counterpart compound BSTS TIs. The Se ordering geometry was investigated by combining state-of-the-art scanning transmission electron microscopy and powder X-ray diffractometry. We demonstrated the existence of inner Se ordering in PbBi2(Te,Se)4 and also in Pb-BSTS TIs. Quantitative analysis of Se ordering and a qualitative view of atomic non-stoichiometry such as point defects are also presented. Pb-BSTS TIs' Se ordering structure and their large gap nature has the great potential to achieve more bulk insulation than conventional BSTS TIs.

Pb Mα1, Bi Mα1, Sb Lα1, Te Lα1 and Se Kα1 were used as characteristic X-ray for each element.

S2. The validity of Richardson-Lucy deconvolution for STEM images
Here we applied Richardson Lucy deconvolution 1,2,3 for HAADF image to enhance the resolution ( Fig. S2a and S2b). Ishizuka et al. utilized these algorithms to enhance spatial resolution in HAADF 4 . Under appropriate iterative process condition 4 , we can effectively reduce the noise of raw data.  and red) and intensity line color are one to one corresponded. We can see the intensity fluctuation of inner Te/Se(2) columns in raw image (row blue and red). In the raw line profile, however, the intensity line profile is noisy and high intensity Se sites (red, row4 for example) and low ones (blue, row4) are hardly distinguished. In the deconvoluted line profile, however, row 4, 5, and 6 have both high and low sites in the same Se(2) inner sites. This tendency is also visible in raw image, so we can conclude that denoising was conducted in an accurate manner.
We also find such fluctuation in other areas (Fig. S2b, yellow arrows), and seemingly they form 4 weak/high intensity pairs with next neighbor Te/Se(2) site across center Pb column. Also, as for ABF image (Fig. S2c), we can see that the denoising process does not change its nature.

S3. Effect of preferred orientation and antisite defects on pXRD intensity
First, we compare the intensity from the same orientation plane; 003, 006, and 009. These belong to the same (00h) plane (cleavage plane of this compound); therefore, even when the preferred orientation occurs, the intensity ratio of these reflection does not change.
Second, according to the report of Shelimova 5 , the dominant anti-site defects in PbBi2Te4 are VTe, TeBi, BiPb, which yield an n-type carrier. BiPb does not affect the intensity in XRD measurements, and VTe is thought to much fewer and hardly affect. The most effective antisite defect in the pXRD intensity is assumed to be the TeBi antisite defects. From the STM study of the Bi2Te2Se compound, the intermixing of Te and Bi rarely happens and ratio is less than 1% 6 .
It should also be the case for Pb-BTS TI, then we calculated the pXRD of the worst case that affects the 00h intensity most: 10% Te migrate to the Bi sites, and 5% migration, comparing with the no migration case (Fig. S3). The more Te migration occurs, the higher the 006 / 003 value becomes. Considering the inset of Fig. 4b in the main text, this tendency may overestimate the Se ordering (for a perfect crystal). However, because the 006 / 003 value changes at most 3.78/3.08=1.24 times by the intermixing of Te and Bi. Experimental value of the 006 / 003 = 2.17 (Fig. 4b in the main text) should be reduced to 2.17/1.24 = 1.75 at its worst case.
7 Figure S3. The Te/Bi intermixing effect to the 006 / 003 ratio. The powder XRD simulation was conducted by RIETAN-FP 7 . As for an isotropic atomic displacement parameter (B), B is reported to range from 1.0 to 2.0 Å 2 in the compound of PbSb2Te4 and PbBi4Te7 5 , and 0.4 Å 2 in Bi2Te2Se 8 . Considering Pb-BSTS is composed of all heavy atoms, we fix B= 1Å 2 for all atoms in this calculation.