Ab initio crystal structures and relative phase stabilities for the aleksite series, Pb n Bi4Te4S n+2

DFT is used to obtain structural information for seven members of the aleksite homologous series. Relationships between modulation vectors and the d subcell can be mathematically defined, allowing the prediction of crystal parameters for any member of the series, a valuable finding for mineral systematics and classification, and for an expanded understanding of this and other mixed-layer series.


Introduction
Several named minerals and a number of unnamed Bi-Pbtellurosulfide phases were initially grouped together within a homologous series with the common formula Pb n Bi 4 Te 4 S n+2 , where n is homologue number (Cook et al., 2007a).Later, the series was termed the aleksite series after the first named mineral, aleksite (Pb 2 Bi 4 Te 4 S 4 ), with the generalized formula revised to Pb (nÀ1) Bi 2 X n+2 (n = homologue number, X = chalcogen) (Moe ¨lo et al., 2008).Cook et al. (2007a) postulated the existence of a hierarchical series of Pb-Bi-tellurosulfides that can be expanded from the archetypal five-atom tetradymite unit to larger seven-, nine-, 11-atom units, whereas Moe ¨lo et al. (2008) considered the tetradymite (Bi 2 Te 2 S) archetype as a link to layered sulfosalts.A second large group of minerals and unnamed compounds, Bi x X y (X = chalcogen), is also derived from the same archetype, constituting the tetradymite homologous series (Cook et al., 2007b).Ciobanu et al. (2009) affirmed that the two homologous series derived from the tetradymite archetype share structural building principles in agreement with the formulae: (i) chalcogen-rich [S(M p X p+1 )ÁL(M p+1 X p+2 ]; p ! 2)] for the aleksite series, and (ii) bismuth-rich [S 0 (Bi 2k X 3 )ÁL 0 [Bi 2(k+1) X 3 ]) for the tetradymite series.Investigation of compounds from the tetradymite series in the compositional range Bi 2 X 3 -Bi 8 X 3 using transmission electron microscopy (TEM) (Ciobanu et al., 2009) confirmed that these are all n-fold (N = layers in the stacking sequence) superstructures of a rhombohedral subcell with c/3 = d 0 $ 2 A ˚. Electron diffraction patterns show two brightest reflections in the centre of d 0 and are described by two modulations vectors: q = Á c Ã sub (q $ homoatomic interval) and q F = F Á c Ã sub ; q F = (i/N)d 0 * = i Á d Ã N , i = S 0 + L 0 .The same basis for crystal structural modularity attributable to other mixed layer compounds (Amelinckx et al., 1989;Frangis et al., 1990) should extend to the aleksite series since their building modules follow the same accretional principle with the tetradymite compounds.This was demonstrated in a high-angle annular dark-field scanning TEM (HAADF STEM) study of unnamed PbBi 4 Te 4 S 3 (Cook et al., 2019).Based on the polytypism observed, Cook et al. (2019) showed that for a single homologue, n = 1 in this case, the structure could consist of combinations of multiple S and L m modules, at constant p = 2.
Ab initio calculation of structures from the tetradymite series within the compositional interval Bi 2 Te 3 -Bi 8 Te 3 has confirmed crystal structural modularity using the accretional formalism as above (Yao et al., in the press).Moreover, the same study formulated a model combining the modulation parameter and d sub to predict the upper (Bi-rich) end of the tetradymite series.
Study of layered compounds in the system PbTe-Bi 2 Te 3 led to the definition of another homologous series based on units of fixed width: nPbTeÁmBi 2 Te 3 (Shelimova et al., 2004) using a similar approach to the definition of nBi 2 ÁmBi 2 Te 3 for  compounds in the tetradymite series (Shelimova et al., 2000).Following the same ideas, Kuribayashi et al. (2019) discovered and named the third member of the aleksite series, hitachiite (Pb 10 Bi 4 Te 4 S 12 ), and introduced the formula Bi 2 Te 2 SÁnPbS to express homology in the series, an approach distinct from the accretional model described above.
Ab initio calculations of phases across an extended compositional range in a modular series provide an excellent tool for the exploration of modularity, crystal structures, phase stabilities, and the limits of the series.Using density functional theory (DFT) and structure simulations we study seven homologues from the aleksite series covering the compositional range Bi 2 Te 2 S-Pb 12 Bi 4 Te 4 S 14 .Our objectives are: (i) to describe their structures, bonding, structural-chemical modulation and phase stabilities, (ii) build a model for predicting the upper (Bi-rich) limit of the series and (iii) discuss similarities and differences between the aleksite and tetradymite series.
All phases are trigonal, but the space group changes from R " 3 3m (R) to P " 3 3m1 (H) whenever the total number of atoms in the explicit formula is divisible by 3. The total number of layers in each structure is N = N 1 Â 3 for R phases and N = N 1 Â 1 in H.The d sub value is calculated from experimental data using c/N total.In the unnamed phases, the c parameter is calculated assuming an interlayer distance of d 0 $ 2 A ˚. We found that a remains constant at around 4.23 A ˚whereas c shows large variation depending on the space group and the stacking sequence of individual polytypes.Nevertheless, their interlayer distances (d sub = c/N total ) are directly comparable with one another and decrease systematically with increasing Pb and S content.
cutoff energy 600 eV.van der Waals interactions (Te-Te) were included using the method of Grimme et al. (2010).Structures were relaxed with energy convergence of less than 10 À6 eV for each ionic step and forces on each atom are within 0.02 eV A ˚À1 .
In order to obtain the equation of state (EOS) for each structure, relaxations were carried out at different volumes with lattice vectors scaled from 95 to 101%.The relationship between volume and total energy was fitted using the Murnaghan (1944) equation of state: where K 0 and K 0 0 are the bulk modulus and its pressure derivative, V 0 is the equilibrium volume and E 0 is the reference energy.For each structure, the relaxed unit-cell parameters are obtained by calculating structure relaxations at equilibrium volume.
Upon completing the structure relaxations for each phase, we calculate the formation energy (ÁE f ) to evaluate the relative phase stability.Applying a similar approach to that used by Woodcox et al. (2019), we establish a simple relation between ÁE f , the energy for each phase (E phase ) and the energy of single atoms (E Bi , E Te , E Pb and E S ) in equation ( 2): where a, b, c and d represent the number of atoms of Bi, Te, Pb, and S, respectively, within each structure.When ÁE f 0, the phase is considered potentially stable.An alternative approach to establishing the relative stability of a phase considers the energy difference to endmembers (Park et al., 2021), i.e. tetradymite (Bi 2 Te 2 S) and galena (PbS), using equation ( 3): where Total and E PbS Total are the total energies of each mixed phase and N atom = total number of atoms.
Table 3 Fitted equation of state parameters for nine minerals and unnamed phases from the aleksite series.
E 0 is the reference energy, V 0 is the equilibrium volume per atom for each simulated cell, K 0 and K 0 0 are the bulk modulus and its derivative.V 0 is compared with available experimental data from the literature.
Chemical formula Bi  Published experimental data (Table 1) are included for comparison.Ttd is tetradymite, Alk is aleksite, Sdd is saddlebackite, Hit is hitachiite.

Crystal structure relaxation
We adopted the experimental a and c unit-cell parameters in Table 1 as input for the total energy calculations.The same procedure used by Yao et al. (in the press) was applied to obtain the initial atomic coordinates for the N number of atoms in each crystal structure.The z coordinates are at equal intervals of 1/N along c, and the corresponding x, y coordinates are at 1 3 , 2 3 and 0 values repeating for a group of three atoms.To obtain the relaxed structures, we firstly constrain the equilibrium volume for each phase by fitting the total energy volume curves (Fig. 2) using the Murnaghan equation of state with EOS parameters tabulated in Table 3.The calculated V 0 values agree with published data (Table 3) within 3.6% for all available structures.
The final structure parameters are obtained from the DFT calculations at the V 0 values for all phases.The relaxed a and c unit-cell parameters are within 1.5% difference with the published data (Tables 1 and 4).Comparison with experimental data shows a slight overestimation in the a parameter [Fig.3(a)] and a good fit for d sub values [Fig. 3(b)].Both a and d sub parameters show a smooth decreasing trend with increasing PbS across the compositional interval investigated.Notably, the double-module polytypes of aleksite and saddlebackite yield values for a and d sub that are very similar to those of their respective single-module polytypes (Fig. 3).

Crystal structure models
The crystal structure models obtained using the relaxed unit-cell parameters are plotted on the zone axis ½11 " 2 20 to illustrate the incremental increase in width of each structure with addition of Pb and S atoms (Figs. 4 and 5).We note that the building modules are centred onto a slab of S-Pb-S . . .Pb-S flanked on each side by Bi-Te atoms.The increment of the central slab can be expressed as: Pb k S k+1 (k = 0-6) for the six homologues discussed here.The modules are always separated by Te-Te layers (van der Waals gaps).A trigonal PbS structure (PbS R ) obtained by transformation from cubic galena (Noda et al., 1987) is included for comparison (Fig. 4).This shows the atomic arrangement in PbS R is very similar to the central slab in aleksite structures when viewed on the ½11 " 2 20 zone axes.
The models show that the tetradymite unit is no longer preserved as such within the single-module structures, although these are required to form all homologues with n < 2 such as the unnamed n = 1 phase with composition PbBi 4 Te 4 S 3

Figure 5
Crystal structure models (ball and stick on left, atom filling on right) for the relaxed structures of phases where n = 6, 8, 10 and 12 on ½11 " 2 20 zone axis.Layer stacks and their corresponding widths are labelled on the top of each.Atom arrangements (red = Bi, blue = Te, yellow = sulfur, green = Pb) depicting the structure are plotted along the (hkil) planes, i = À(h+k).A crystallographic information file (CIF) is provided in the supporting information.

Table 4
Calculated unit-cell parameters a and c for nine relaxed structures based on density functional theory.
Values for the interlayer distance d sub are obtained from the c unit-cell parameter and total number of layers (N total) for each phase.

Name
Formula n Space group a  (Cook et al., 2019).Additionally, 5-atom tetradymite modules are constituent building blocks in the double-module aleksite polytype, aleksite-42R (n = 2) considered in this contribution, which contains a (5.9) sequence (Fig. 4).Five-atom tetradymite modules may potentially exist in other configurations with n > 2 (e.g.5.13 saddlebackite), although these will not be considered here.Alongside single-module 9H saddlebackite (n = 4), we do, however, consider the 18H double-module polytype structure of saddlebackite, which features a (7.11) sequence.We note that the simplest polytypes representing the n = 2k+2 (k = integer) building modules are composed of two single-module units with n = 2k and n = 2k + 4. Slight lattice distortion is observed within structures containing two different modules, e.g. the shift between the 5and 9-atom layer stacks in the aleksite-42R polytype.Variations in atom arrangements within the single-module structures are, in contrast, negligible.

Bond analysis
The bond types and their variation in length across the studied phases are shown within the asymmetric unit cell for all structures (Figs. 6 and 7).The three types of bonds in tetradymite (Bi-Te, Bi-S and Te-Te) are complemented by Pb-S bonds in all other compounds from the aleksite series.The Bi-Te bond decreases in length from tetradymite (3.047A ˚) to the 17-atom layer structure (3.036A ˚) whereas the length of the Bi-S bond remains relatively constant at $3.019 A ˚. Likewise, Te-Te bond lengths increase from 3.882 A ˚in tetradymite to a maximum of 3.919 A ˚in the 11atom layer.The Te-Te bonds in the two aleksite polytypes are constant and close to those in tetradymite (3.884A ˚and 3.886 A ˚for the 5.9-and 7.11-atom layer sequences, respectively).Although the average Pb-S bond length is nearly constant $2.990 A ˚, there is a small variation within the middle Pb k S k+1 slab, e.g. from 2.986 to 2.995 A ˚in the 17-atom layer.The Pb-S bond lengths within the aleksite and saddlebackite double-module polytypes are nearly identical to those in the corresponding single-module polytypes.
In Figs. 6 and 7, bond lengths are projected onto the c axis to calculate the contribution towards the d sub value in each structure.Te-Te bond projections have the highest values on the c axis whereas Bi-Te and Bi-S projections are only slightly larger than Pb-S bond projections.In all single-module structures, there is one Te-Te bond, two Bi-Te bonds and two Bi-S bonds, whereas the number of Pb-S bonds increases from 0 in tetradymite to 12 in the 17-atom layer, with an incremental step of 2. We thus divide the bond types into two groups: variable number (Pb-S) and fixed number (Bi-Te, Bi-S and Te-Te).
We have calculated the bond contribution to the d sub parameter from cumulative projection values and their abundance across the compositional range studied (Fig. 8).This plot shows two opposing trends, an increase in Pb-S contribution and decrease in contribution from other bonds from tetradymite to the 17-atom layer, the two lines inter-

STEM simulation and electron diffraction
In Figs. 9 and 10 we show the relaxed structures in STEM simulations and electron diffraction (ED) patterns on zone axis ½11 " 2 20.The signal intensity (I) in (HAADF) STEM imaging is proportional to Z 2 of an element along an atomic column (Pennycook & Jesson, 1990;Ishizuka, 2002).We note that S (Z = 16) is not displayed on the simulations when the neighbouring atom columns are Bi (Z = 83) and Pb (Z = 82).A better visualization of the number of atoms in each structure can be assembled using Se (Z = 34) instead of S (inset, top right in Fig. 9 and overlays on each STEM simulation in Fig. 10).Such simulations agree very well with the atomic arrangement models displayed in Figs. 4 and 5 and are concordant with HAADF STEM images of phases from the aleksite series (Cook et al., 2019).
The d Ã sub interval (cropped from the ED patterns) is essential for constraining structural modulation in terms of the increase in module width and module combinations.Within each d Ã sub interval, there are N 1 À 1 number of reflections equally distributed.The asymmetric unit-cell length (d N 1 ) is correlated with layer stacks for each structure.This can be also indicated as the smallest interval, d Ã N 1 between two neighbour reflections along d Ã sub .Typical of all phases in the series is the fact that the ED patterns show the two brightest reflections at the centre of d Ã sub .This interval, underpinned by the modulation vector q F = F Á d Ã sub , where F = i/N 1 Ád sub $ 1/N 1 ; i = S + L. F values are within the range 0.2-0.059for the analysed structures and this shows a monotonic decrease with increase in PbS concentration.In cases where there are multiple polytypes, although the number of divisions is doubled relative to their single unit structure, e.g. 14 and 7 divisions for the 42R and 21R aleksite polytypes, the q F vector remains unchanged.Nonetheless, the q F interval is split into two by a satellite reflection of lesser intensity (Fig. 9).On the other hand, the displacive modulation between chalcogen (S, Te, Se) and Pb and Bi atoms is underpinned by a second vector: q = Á c Ã sub (Lind & Lidin, 2003).The q modulation is depicted up to third-order reflections along c* (ED patterns in Figs. 9 and 10).Values of (1.8-1.588 for the 5-to 17-atom layer module range) are calculated as 3[(N 1 + 1)/2]/N 1 for single modules.In the double-module polytypes = 3[(N 1 + 2)/2]/N 1 giving the same values of as the corresponding single-module structures (Fig. 9).
This formalism is in agreement with the crystal structural formula: S(M p X p+1 )ÁL(M p+1 X p+2 ), X = chalcogen, where S and L are the number of shorter and longer modules (Cook et al., 2019) but not the formula nPbTeÁmBi 2 Te 3 of Shelimova et al. (2004).For example, aleksite-21R would have n = m = 1, requiring two distinct modules instead of only one.The 42R polytype will have 4 modules (n = m = 2) instead of the '5' and '9' modules considered here.Such a strong correlation between electron diffraction patterns and chemical modules in a homologous series is typical for mixed layer compounds (Amelinckx et al., 1989).

Phase stability and energy mixing
Formation energies [equation ( 2)] for the studied phases in the interval Bi 2 Te 2 S-Pb 12 Bi 4 Te 4 S 14 (n = 12) are given in Table 5. Calculation of the formation energy and energy of mixing requires the DFT reference energies (E 0 ) of all elements (Bi, Pb, Te and S) and endmembers (Bi 2 Te 2 S and PbS).The reference energies for Pb, S, Bi 2 Te 2 S (Table 3), and PbS are calculated from equation of state fitting [equation ( 1)] in this study, those for the elements Bi and Te are adopted from Yao et al. (in the press).Their parameters are summarized in Table 6.All reference energies are calculated based on the GGA functional.
The calculated ÁE f values are negative for all nine phases and decrease as the PbS component increases, implying they are relative stable to the endmembers.The larger doublemodule polytypes of both aleksite (42R) and saddlebackite (18H) show the same formation energy as their corresponding single-module units (21R and 9H, respectively), implying they are equally stable.
Phase stability can also be evaluated from the energy of mixing (E mixing ), which is calculated using values of the two endmember phases, tetradymite and galena [equation (3)]; Table 5).This defines a convex hull between tetradymite and galena (PbS) with aleksite at the lowest energy point (Fig. 11).The other five studied homologues plot along or slightly below the branch between aleksite and galena.Such a distribution indicates that all studied phases can be relative stable compared with the endmembers and thus do not readily decompose into tetradymite and galena endmembers.However, whether the studied phases are thermodynamically stable may require further phonon calculations to investigate the thermal effects and entropy contributions (e.g.Belmonte et al., 2014).
Instead of adopting the formula nPbSÁmBi 2 Te 2 S as a working model, the energy of mixing can also be defined using the accretional model: where S = 5-atom layer, L 1-3 represent longer 7-, 9-and 11modules; M = Bi, Pb, and X = Te, S. The energy of mixing for aleksite-42R (5.9) and saddlebackite-18H (7.11) are found at 0 and 0.2 meV per atom, respectively.This shows ideal mixing when using the accretional model and indicates that the derived polytypes and, indeed, other multiple-module structures in the series can be stable relative to their single-module components.Further calculations may, however, be required to fully validate these findings.Preliminary work shows that homologues of the aleksite series with still greater PbS content (n = 18 and n = 30, representing 23-and 35-atom layers, respectively) are present in assemblages buffered by galena (Cook et al., 2021 and unpublished data).Theoretical phases from the PbS-rich end of the series, such as 403-, 205-, 71-and 51-atom layers (corresponding to homologues with n = 398, 200, 66 and 46), can also be considered based on their chemistry, which is close to, but distinct from, PbS (Fig. 1).
Our model describes a quasilinear relationship between and d sub (Fig. 12), which allows the prediction of d sub for any phase across the 17-to 403-atom layer structure range ( = 1.588-1.504),with d sub values over this interval lying in the range 1.806 to 1.726 A ˚.The theoretical 403-atom layer phase, Pb 398 Bi 4 Te 4 S 400 , with Pb/(Pb + Bi) = 0.99 shows identical d sub values as our DFT-modelled predictions for PbS T , which is also within 0.8% difference of that for trigonal PbS transformed from the experimental cubic structure (Noda et al., 1987).As a result, our model is suitable to approximate d sub values for aleksite series homologues across the entire compositional range from tetradymite to the PbS T endmember.4. The ribbon below each STEM image is cropped from the ED patterns, showing the number of reflections and their intensity variations along the d Ã sub interval.Two modulation vectors (q and q F ) underpinning structural modulation are marked by arrows.The atom layer arrangement for each structure is marked by circles (cyan = Te, red = Bi, yellow = S and green = Pb).Layer stacks within all structures are marked by the overlays on the images.Note that simulations for hitachiite (15H) were carried out with space group P1 rather than P " 3 3m1.Table 6 Fitted equation of state parameters for four elements (Pb, S, Bi, Te) and endmember PbS.
E 0 represents the reference energy, V 0 is the equilibrium volume per atom for each simulated cell, K 0 and K 0 0 are the bulk modulus and its derivative, respectively.V 0 and K 0 are compared with the available experimental data.

Modularity and comparison with the tetradymite series
Noting the possibility of multiple polytypes for many, if not all, homologues in the aleksite series (Cook et al., 2019), we introduce a modified formula: where S represents the number of 5-atom layers, and L 1 , L 2 , . . .L m are the numbers of longer, 7-, 9-, . . .2m + 5 modules; m > 0, integer; and S, L ! 0. This formula is useful for expressing the range of polytypes for each homologue within the series.Therefore, applying formula ( 6 The theory of mixed layer compounds stipulates that structures built by modules which are distinct in size and chemical composition are related to one another by characteristics of electron diffraction patterns thus underpinning the modularity within a homologous series (Amelinckx et al., 1989).Both the aleksite and tetradymite series are formed by modular structures derived from the same 5-atom archetype but with distinct compositional ranges, i.e. extending towards PbS (aleksite series) and Bi endmembers (tetradymite series).
The individual building modules in each series are composed of an uneven number of atoms, 7, 9, 11 . . . 2 k+ 1, but with different topology between cations (Bi, Pb) and chalcogens (Te, S, Se), i.e., symmetrical in the aleksite series and asymmetrical in the tetradymite series.Despite this, the electron diffractions of relaxed structures from the aleksite series (Figs. 9 and 10) show identical modulation vectors as corresponding phases in the tetradymite series with the same number of layers and/or building modules (Ciobanu et al., 2009;Yao et al., in the press).Such characteristics provide a strong link between the two series and prove their affiliation to a single class of mixed layer compounds built by the same accretional homology principles.The alternative homology proposed for the two series involving units of the same size, 2and 5-atom layers (Shelimova et al., 2000(Shelimova et al., , 2004;;Kuribayashi et al., 2019) is not supported by the crystal structures, even though it may be conceptually useful to depict chemical variation within each of the two series.
Bond analysis shows marked differences between the two series, whereby the longer Te-Te bonds are present in all homologues of the aleksite series and may be responsible for the extensive polytypism.In contrast, the Te-Te bonds are only present in Te-rich members of the tetradymite series (Yao et al., in the press).Construction of incremental symmetrical modules by addition of Pb-S in the aleksite series and by asymmetrical modules involving Bi-Bi pairs in the tetradymite series leads to linear versus non-linear features in the respective -d sub relationships.As a result, for the same interval (1.8-1.5) the range of d sub is greater for the aleksite series compared with the tetradymite series, i.e. $2 to 1.726 A ˚, and $2 to 1.973 A ˚, respectively (Fig. 12).

Conclusions and implications
The crystal structures and stabilities of phases from the aleksite homologous series, Pb n Bi 4 Te 4 S n+2 , where n = homologue number (Cook et al., 2019), were calculated using DFT methods.The study addressed four named minerals (tetradymite, aleksite, saddlebackite and hitachiite) and three compounds yet to be described in natural specimens

Figure 1
Figure 1 Plot of Pb/(Pb+Bi) versus Te/(Te+S) indicating compositions of phases in the aleksite series.The seven single-module structures (5-, 7-, 9-, 11-, 13-, 15and 17-atom layers) are indicated by large black stars and the two corresponding double-layer polytypes 5.9 (7-), 7.11 (9-) are represented by filled red stars.Theoretical phases close to the PbS end of the series with 51-, 71-, 205-and 403-atom layers are shown by grey stars.The latter are also plotted together with galena (blue star) in the inset figure, for clarity.

Figure 2
Figure 2Total energy curves of the nine structures corresponding to seven phases in the aleksite series as a function of the atomic volume.Open black circles represent DFT-calculated total energies.Solid black lines are determined by fitting the Murnaghan equation of state (1).Seven of these are single-module structures n = 0, 2, 4, 6, 8, 10, 12, and two are the double-layer polytypes of aleksite (5.9) and saddlebackite (7.11).

Figure 3
Figure 3 Variation in (a) unit-cell parameter a and (b) interlayer distance d sub plotted as a function of PbS/(PbS+Bi 2 Te 2 S) for the seven studied phases.Published experimental data (Table1) are included for comparison.Ttd is tetradymite, Alk is aleksite, Sdd is saddlebackite, Hit is hitachiite.

Figure 4
Figure 4Crystal structure models (ball and stick on left, atom filling on right) for the relaxed structures of phases where n = 0, 2 and 4 on ½11 " 2 20 zone axis.Trigonal PbS from literature is also shown.Layer stacks and their corresponding widths are labelled on the top of each.Atom arrangements (red = Bi, blue = Te, yellow = S and green = Pb) depicting the structure are plotted along the (hkil) planes, i = À(h+k).A crystallographic information file (CIF) is provided in the supporting information.

Figure 6
Figure 6 Bond types and bond lengths for the relaxed structures of phases where n = 0, 2 and 4 within the asymmetric unit cell plotted on the ½11 " 2 20 zone axis.Trigonal PbS from the literature is also illustrated for comparison.Red = Bi, blue = Te, yellow = S, green = Pb.Projections of bonds along the c axis are labelled underneath, they represent the bond length contributions to d sub shown in Fig. 8.

Figure 7
Figure 7 Bond types and lengths for the relaxed structures of phases where n = 6, 8, 10 and 12 within the asymmetric unit cell plotted on ½11 " 2 20 zone axis.Red = Bi, blue = Te, yellow = sulfur, green = Pb.Projections of bonds along the c axis are labelled on the bottom.They represent the bond length contributions to d sub shown in Fig. 8.

Figure 8
Figure 8 Bonds length contributions to d sub for the seven single-module phases (n = 0, 2, 4, 6, 8, 10 and 12).The Pb-S contribution is shown in blue, and others (Bi-Te, Bi-S and Te-Te) are shown in orange.The two types of bonds display contrasting trends: the contribution from Pb-S increasing from the 5-to 17-atom layer, whereas the contribution of other bonds systematically decreases.

Figure 9
Figure 9 STEM simulations (left) and corresponding electron diffraction (ED) patterns (right) for the relaxed structures of tetradymite (15R), aleksite (21R and 42R) and saddlebackite (9H and 18H) shown on the ½11 " 2 20 zone axis.Unit-cell parameters a and c and the interlayer distance d sub for each structure are tabulated Table4.The ribbon below each STEM image is cropped from the ED patterns, showing the number of reflections and their intensity variations along the d Ã sub interval.Two modulation vectors (q and q F ) underpinning structural modulation are marked by arrows.The atom layer arrangement for each structure is marked by circles (cyan = Te, red = Bi, yellow = S and green = Pb).Layer stacks within all structures are placed top right on the figure.Note that simulations for saddlebackite (9H and 18H) were carried out with space group P1 rather than P " 3 3m1.

Figure 10 STEM
Figure 10 STEM simulations (left) and corresponding electron diffraction (ED) patterns (right) for the relaxed structures of Pb 6 Bi 4 Te4S 8 (33R), Pb 8 Bi 4 Te 4 S 10 (39R), hitachiite (15H) and Pb 12 Bi 4 Te 4 S 14 (51R) shown on the ½11 " 2 20 zone axis.Unit-cell parameters a and c and the interlayer distance d sub for each structure are tabulated in Table4.The ribbon below each STEM image is cropped from the ED patterns, showing the number of reflections and their intensity variations along the d Ã sub interval.Two modulation vectors (q and q F ) underpinning structural modulation are marked by arrows.The atom layer arrangement for each structure is marked by circles (cyan = Te, red = Bi, yellow = S and green = Pb).Layer stacks within all structures are marked by the overlays on the images.Note that simulations for hitachiite (15H) were carried out with space group P1 rather than P " 3 3m1.

Figure 11
Figure 11 Phase stability of aleksite series phases evaluated from a plot of energy of mixing versus PbS/(PbS+Bi 2 Te 2 S).The plot features a convex Ttd-Alk-PbS hull.The phases Sdd, Pb 6 Bi 4 Te 4 S 8 , Pb 8 Bi 4 Te 4 S 10 , Pb 10 Bi 4 Te 4 S 12 , Hit and Pb 12 Bi 4 Te 4 S 14 all lie below the Alk-PbS part of hull, indicating they can be stable.The two endmembers, tetradymite and PbS, are represented by blue circles; the seven aleksite series phases are represented by red circles.Ttd = tetradymite, Alk = aleksite, Sdd = saddlebackite, Hit = hitachiite.

Figure 12
Figure 12 Model of the relation between and d sub for the aleksite series (red circles, solid blue line, this study) compared with the tetradymite series (green circles, solid black line) from Yao et al. (in the press).The linear curve for the aleksite series is fitted from the and d sub values of the seven single-module phases.This allows extrapolation to theoretical phases (purple circles) the end of the series near PbS.Phases with the same values from the two series are highlighted by dashed lines.

Table 1
Background and crystal structure information for studied phases from the aleksite series.

Table 2
Number of atoms and chemical formula units and k points employed in each simulation box for all nine aleksite series structures.

Table 5
DFT calculated formation energy (E f ) and energy of layer mixing (E mixing ) for the nine phases from the aleksite series.
Corresponding equations (2) and (3) are given in the text.