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Ab initio crystal structures and relative phase stabilities for the aleksite series, PbnBi4Te4Sn+2
aSchool of Chemical Engineering, The University of Adelaide, North Terrace, Adelaide, South Australia 5005, Australia, and bBHP Olympic Dam, 10 Franklin Street, Adelaide, S.A. 5000, Australia
*Correspondence e-mail: jie.yao@adelaide.edu.au
Density functional theory methods are applied to crystal structures and stabilities of phases from the aleksite homologous series, PbnBi4Te4Sn+2 (n = homologue number). The seven phases investigated correspond to n = 0 (tetradymite), 2 (aleksite-21R and -42R), 4 (saddlebackite-9H and -18H), 6 (unnamed Pb6Bi4Te4S8), 8 (unnamed Pb8Bi4Te4S10), 10 (hitachiite) and 12 (unnamed Pb12Bi4Te4S14). These seven phases correspond to nine single- or double-module structures, each comprising an odd number of atom layers, 5, 7, (5.9), 9, (7.11), 11, 13, 15 and 17, expressed by the formula: S(MpXp+1)·L(Mp+1Xp+2), where M = Pb, Bi and X = Te, S, p ≥ 2, and S and L = number of short and long modules, respectively. Relaxed structures show a and c values within 1.5% of experimental data; a and the interlayer distance dsub decrease with increasing PbS content. Variable Pb—S bond lengths contrast with constant Pb—S bond lengths in galena. All phases are n-fold superstructures of a rhombohedral with c/3 = dsub*. Electron diffraction patterns show two brightest reflections at the centre of dsub*, described by the modulation vector qF = (i/N) · dsub*, i = S + L. A second modulation vector, q = γ · csub*, shows a decrease in γ, from 1.8 to 1.588, across the n = 0 to n = 12 interval. The linear relationship between γ and dsub allows the prediction of any theoretical phases beyond the studied compositional range. The upper PbS-rich limit of the series is postulated as n = 398 (Pb398Bi4Te4S400), a phase with dsub (1.726 Å) identical to that of trigonal PbS within experimental error. The aleksite series is a prime example of mixed layer compounds built with accretional homology principles.
1. Introduction
Several named minerals and a number of unnamed Bi–Pb–tellurosulfide phases were initially grouped together within a homologous series with the common formula PbnBi4Te4Sn+2, where n is homologue number (Cook et al., 2007a). Later, the series was termed the aleksite series after the first named mineral, aleksite (Pb2Bi4Te4S4), with the generalized formula revised to Pb(n−1)Bi2Xn+2 (n = homologue number, X = chalcogen) (Moëlo 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 Moëlo et al. (2008
) considered the tetradymite (Bi2Te2S) archetype as a link to layered sulfosalts. A second large group of minerals and unnamed compounds, BixXy (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(MpXp+1)·L(Mp+1Xp+2]; p ≥ 2)] for the aleksite series, and (ii) bismuth-rich [S′(Bi2kX3)·L′[Bi2(k+1)X3]) for the tetradymite series. Investigation of compounds from the tetradymite series in the compositional range Bi2X3–Bi8X3 using (TEM) (Ciobanu et al., 2009
) confirmed that these are all n-fold (N = layers in the stacking sequence) superstructures of a rhombohedral with c/3 = d0 ∼ 2 Å. Electron diffraction patterns show two brightest reflections in the centre of d0 and are described by two modulations vectors: q =
(q ∼ homoatomic interval) and qF =
; qF = (i/N)d0* =
, i = S′ + L′.
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 PbBi4Te4S3 (Cook et al., 2019
). Based on the 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 Lm modules, at constant p = 2.
Ab initio calculation of structures from the tetradymite series within the compositional interval Bi2Te3–Bi8Te3 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 γ and dsub to predict the upper (Bi-rich) end of the tetradymite series.
Study of layered compounds in the system PbTe–Bi2Te3 led to the definition of another homologous series based on units of fixed width: nPbTe·mBi2Te3 (Shelimova et al., 2004) using a similar approach to the definition of nBi2·mBi2Te3 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 (Pb10Bi4Te4S12), and introduced the formula Bi2Te2S·nPbS 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 Bi2Te2S–Pb12Bi4Te4S14. 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.
2. data and selection of input files
Table 1 lists the seven phases under investigation (four minerals and three unnamed phases) and published data relating to their crystal structures. These are also shown on a diagram of (Pb/Pb+Bi) versus Te/(Te+S) (Fig. 1
). They represent seven discrete homologues with even-numbered values of n (0, 2, 4, 6, 8, 10 and 12) using the formula PbnBi4Te4Sn+2 given by Cook et al. (2007a
, 2019
). The corresponding structures are given by the formula S(MpXp+1)·L(Mp+1Xp+2), where p = 2, S = the 5-atom layer (for simplicity termed `5-layer' hereafter), and six different L modules (7-, 9-, 11-, 13-, 15- and 17-layers). In addition, homologues n = 2 and n = 4, corresponding to the minerals aleksite (7-layer) and saddlebackite (9-layer), respectively, are each represented by simple, double-module [aleksite-42R with the stacking sequence (5.9), and saddlebackite-18H with (7.11) stacking sequence]. The aleksite-54H polytype (Spiridonov, 1995
), with much longer stacking sequence (77.11.77.15), was not included due to the much longer computation time required.
|
![]() | 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-, 15- and 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. |
All phases are trigonal, but the (R) to
(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 = N1 × 3 for R phases and N = N1 × 1 in H. The dsub value is calculated from experimental data using c/Ntotal. In the unnamed phases, the c parameter is calculated assuming an interlayer distance of d0 ∼ 2 Å. We found that a remains constant at around 4.23 Å whereas c shows large variation depending on the and the stacking sequence of individual Nevertheless, their interlayer distances (dsub = c/Ntotal) are directly comparable with one another and decrease systematically with increasing Pb and S content.
3. Methods
3.1. Ab initio calculations
To understand the connection between ab initio total energy calculations and structure relaxations based on density functional theory (DFT) (Hohenberg & Kohn, 1964; Kohn & Sham, 1965
). We used the VASP simulation package (Kresse & Furthmüller, 1996
) based on the projector augmented wave (PAW) method (Blöchl, 1994
). The exchange and are treated with the generalized gradient approximation (GGA) within the Perdew, Burke and Ernzerhof (PBE) scheme (Perdew et al., 2008
). The gamma-centered dense k points were used to sample the (Table 2
) and plane waves are expanded at 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 10−6 eV for each ionic step and forces on each atom are within 0.02 eV Å−1.
|
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 K0 and are the bulk modulus and its pressure derivative, V0 is the equilibrium volume and E0 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 (ΔEf) to evaluate the relative phase stability. Applying a similar approach to that used by Woodcox et al. (2019), we establish a simple relation between ΔEf, the energy for each phase (Ephase) and the energy of single atoms (EBi, ETe, EPb and ES) 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 ΔEf ≤ 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 (Bi2Te2S) and galena (PbS), using equation (3)
:
where ,
and
are the total energies of each mixed phase and Natom = total number of atoms.
4. Results
4.1. 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 The z coordinates are at equal intervals of 1/N along c, and the corresponding x, y coordinates are at
,
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 V0 values agree with published data (Table 3
) within 3.6% for all available structures.
|
![]() | Figure 2 Total 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) ![]() |
The final structure parameters are obtained from the DFT calculations at the V0 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 dsub values [Fig. 3
(b)]. Both a and dsub parameters show a smooth decreasing trend with increasing PbS across the compositional interval investigated. Notably, the double-module of aleksite and saddlebackite yield values for a and dsub that are very similar to those of their respective single-module (Fig. 3
).
|
![]() | Figure 3 Variation in (a) unit-cell parameter a and (b) interlayer distance dsub plotted as a function of PbS/(PbS+Bi2Te2S) for the seven studied phases. Published experimental data (Table 1 ![]() |
4.2. models
The 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: PbkSk+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 (PbSR) obtained by transformation from cubic galena (Noda et al., 1987
) is included for comparison (Fig. 4
). This shows the atomic arrangement in PbSR is very similar to the central slab in aleksite structures when viewed on the
zone axes.
![]() | Figure 4 Crystal structure models (ball and stick on left, atom filling on right) for the relaxed structures of phases where n = 0, 2 and 4 on |
![]() | 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 |
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 PbBi4Te4S3 (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 representing the n = 2k+2 (k = integer) building modules are composed of two single-module units with n = 2k and n = 2k + 4.
Slight e.g. the shift between the 5- and 9-atom layer stacks in the aleksite-42R polytype. Variations in atom arrangements within the single-module structures are, in contrast, negligible.
is observed within structures containing two different modules,4.3. Bond analysis
The bond types and their variation in length across the studied phases are shown within the 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 (3.047 Å) to the 17-atom layer structure (3.036 Å) whereas the length of the Bi—S bond remains relatively constant at ∼3.019 Å. Likewise, Te—Te bond lengths increase from 3.882 Å in tetradymite to a maximum of 3.919 Å in the 11-atom layer. The Te—Te bonds in the two aleksite are constant and close to those in tetradymite (3.884 Å and 3.886 Å for the 5.9- and 7.11-atom layer sequences, respectively). Although the average Pb—S bond length is nearly constant ∼2.990 Å, there is a small variation within the middle PbkSk+1 slab, e.g. from 2.986 to 2.995 Å in the 17-atom layer. The Pb—S bond lengths within the aleksite and saddlebackite double-module are nearly identical to those in the corresponding single-module polytypes.
![]() | Figure 6 Bond types and bond lengths for the relaxed structures of phases where n = 0, 2 and 4 within the cell plotted on the ![]() |
![]() | Figure 7 Bond types and lengths for the relaxed structures of phases where n = 6, 8, 10 and 12 within the cell plotted on ![]() |
In Figs. 6 and 7
, bond lengths are projected onto the c axis to calculate the contribution towards the dsub 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 dsub 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 intersecting at the 11-atom layer. The increase in the contribution to dsub from the Pb—S bonds is however more moderate than the contribution decreases from the other bonds, resulting in a modest decrease of dsub with increased PbS concentration [Fig. 3
(b)].
![]() | Figure 8 Bonds length contributions to dsub 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. |
5.4. STEM simulation and electron diffraction
In Figs. 9 and 10
we show the relaxed structures in STEM simulations and electron diffraction (ED) patterns on
. The signal intensity (I) in (HAADF) STEM imaging is proportional to Z2 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
).
![]() | 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 ![]() |
![]() | Figure 10 STEM simulations (left) and corresponding electron diffraction (ED) patterns (right) for the relaxed structures of Pb6Bi4Te4S8 (33R), Pb8Bi4Te4S10 (39R), hitachiite (15H) and Pb12Bi4Te4S14 (51R) shown on the ![]() |
The 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
interval, there are N1 − 1 number of reflections equally distributed. The asymmetric unit-cell length ( dN1) is correlated with layer stacks for each structure. This can be also indicated as the smallest interval, dN1* between two neighbour reflections along
.
Typical of all phases in the series is the fact that the ED patterns show the two brightest reflections at the centre of . This interval, underpinned by the modulation vector qF =
, where γF = i/N1·dsub ∼ 1/N1; i = S + L. γF values are within the range 0.2–0.059 for the analysed structures and this shows a monotonic decrease with increase in PbS concentration. In cases where there are multiple 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 the qF vector remains unchanged. Nonetheless, the qF interval is split into two by a satellite reflection of lesser intensity (Fig. 9
). On the other hand, the between chalcogen (S, Te, Se) and Pb and Bi atoms is underpinned by a second vector: q =
(Lind & Lidin, 2003
). The q modulation is depicted up to third-order reflections along c* (ED patterns in Figs. 9
and 10
). Values of γ (1.8–1.588 for the 5- to 17-atom layer module range) are calculated as 3[(N1 + 1)/2]/N1 for single modules. In the double-module γ = 3[(N1 + 2)/2]/N1 giving the same values of γ as the corresponding single-module structures (Fig. 9
).
This formalism is in agreement with the crystal S(MpXp+1)·L(Mp+1Xp+2), X = chalcogen, where S and L are the number of shorter and longer modules (Cook et al., 2019) but not the formula nPbTe·mBi2Te3 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
).
5. Discussion
5.1. Phase stability and energy mixing
Formation energies [equation (2)] for the studied phases in the interval Bi2Te2S–Pb12Bi4Te4S14 (n = 12) are given in Table 5
. Calculation of the formation energy and energy of mixing requires the DFT reference energies (E0) of all elements (Bi, Pb, Te and S) and endmembers (Bi2Te2S and PbS). The reference energies for Pb, S, Bi2Te2S (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 ΔEf values are negative for all nine phases and decrease as the PbS component increases, implying they are relative stable to the endmembers. The larger double-module 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 (Emixing), 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 contributions (e.g. Belmonte et al., 2014
).
![]() | Figure 11 Phase stability of aleksite series phases evaluated from a plot of energy of mixing versus PbS/(PbS+Bi2Te2S). The plot features a convex Ttd–Alk–PbS hull. The phases Sdd, Pb6Bi4Te4S8, Pb8Bi4Te4S10, Pb10Bi4Te4S12, Hit and Pb12Bi4Te4S14 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. |
Instead of adopting the formula nPbS·mBi2Te2S as a working model, the energy of mixing can also be defined using the accretional model:
where S = 5-atom layer, L1–3 represent longer 7-, 9- and 11-modules; 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 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.
5.2. The γ–dsub relationship: a model for the extension of the aleksite series
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 γ and dsub (Fig. 12), which allows the prediction of dsub for any phase across the 17- to 403-atom layer structure range (γ = 1.588–1.504), with dsub values over this interval lying in the range 1.806 to 1.726 Å. The theoretical 403-atom layer phase, Pb398Bi4Te4S400, with Pb/(Pb + Bi) = 0.99 shows identical dsub values as our DFT-modelled predictions for PbST, 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 dsub values for aleksite series homologues across the entire compositional range from tetradymite to the PbST endmember.
![]() | Figure 12 Model of the relation between γ and dsub 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. ![]() |
5.3. Modularity and comparison with the tetradymite series
Noting the possibility of multiple et al., 2019), we introduce a modified formula:
where S represents the number of 5-atom layers, and L1, L2,…Lm 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 for each homologue within the series. Therefore, applying formula (6) to PbnBi4Te4Sn+2 from Cook et al. (2007a
, 2019
), we can calculate the homologue number (n) by relating the total number of cations and chalcogens within the component modules:
n + 4 = 2S + 3L1 +…(m + 2)Lm, leading to n = 2S + 3L1 +…(m + 2)Lm − 4 for the number of cations, and n + 6 = 3S + 4L1 +…(m + 3)Lm, leading to n = 3S + 4L1 +…(m + 3)Lm − 6 for the number of chalcogens.
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, 2- and 5-atom layers (Shelimova et al., 2000
, 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 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 γ–dsub relationships. As a result, for the same γ interval (1.8–1.5) the range of dsub is greater for the aleksite series compared with the tetradymite series, i.e. ∼2 to 1.726 Å, and ∼2 to 1.973 Å, respectively (Fig. 12
).
6. Conclusions and implications
The crystal structures and stabilities of phases from the aleksite homologous series, PbnBi4Te4Sn+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 (Pb6Bi4Te4S8, Pb8Bi4Te4S10 and Pb12Bi4Te4S14). The seven phases represent homologues where n = 0, 2, 4, 6, 8, 10 and 12. Each homologue corresponds to a single-module type with an uneven number of atoms (5, 7, 9, 11, 13, 15 and 17, respectively), expressed by the formula: S(MpXp+1)·L(Mp+1Xp+2), where M = Pb, Bi, and X = Te, S), p ≥ 2, S = five-atom layer, and L1–6 = 7, 9, 11, 13, 15 and 17-atom layers. The n = 2 and n = 4 homologues are also represented by two-layer (aleksite-42R and saddlebackite-18H), which have structures comprising two differently sized modules, (5.9) and (7.11), respectively. Other multi-layer are predicted to exist for phases across the series.
The relaxed structures show the unit-cell parameters a and c within 1.5% of available experimental data. Both a and the interlayer distance dsub show decrease with increasing PbS component in the relaxed structures. models and STEM simulations show that the six single modules (for structures with n > 0) are centred onto a PbkSk+1 slab (k = 1–6), with S–Pb–S…Pb–S arrangement flanked by Bi–Te atoms. We show variable Pb—S bond lengths in the aleksite homologues, representing an important structural difference compared to the constant Pb—S bond lengths in galena.
Electron diffraction patterns show N1 intervals of equal length along demonstrating that all phases are n-fold superstructures of a rhombohedral with c/3 =
. The modulation vector q =
shows a decrease in γ, from 1.8 to 1.588, with increasing PbS component across the compositional range studied (n = 0 to 12). The ED patterns have two brightest reflections at the centre of
, which are described by the modulation vector qF =
(γF = 0.2–0.059). The number of divisions within this central interval corresponds to the number of modules, i.e. 1 for single, and 2 for double modules. This result proves that the homologous structures can be described by the formula S(MpXp+1)·L(Mp+1Xp+2), and not the formula nPbS·mBi2Te2S, involving 2- and 5-atom building units (Shelimova et al., 2000
).
The DFT method is also used to obtain the formation energies and energy of mixing for all seven compositions. The seven single-module structures and the two double-module
show negative formation energies, implying they can be relative stable to their endmembers.We established a linear γ and dsub model which allows the calculation of dsub for any phase beyond the compositional range studied, e.g. phases with n values of 46, 66, 200 and 398. The model predicted a dsub value of 1.726 Å for the phase Pb398Bi4Te4S400 (n = 398). This can be considered as the upper end of the series, as this is the same value obtained for PbST in DFT calculations, and lies within 0.8% of experimental data.
The aleksite and tetradymite series represent excellent examples of mixed layer compounds built by accretional homology principles derived from a shared 5-atom layer archetype. This study illustrates how DFT calculations can not only support predictive models for crystal and chemical modularity, but also represent a tool to expand and ultimately constrain the limits of modular series. Potential applications exist to model other, chemically different, mixed layer structures.
Supporting information
Bi2Te2S | c = 30.02341 Å |
Mr = 705.23 | V = 477.50 Å3 |
Trigonal, R3m | Z = 3 |
a = 4.28540 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.00000 | 0.00000 | 0.10902 | ||
Te1 | 0.00000 | 0.00000 | 0.71648 | ||
S1 | 0.00000 | 0.00000 | 0.50000 |
Pb2Bi4Te4S4 | c = 40.41287 Å |
Mr = 1888.98 | V = 638.16 Å3 |
Trigonal, R3m | Z = 3 |
a = 4.27010 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.00000 | 0.00000 | 0.58139 | ||
Pb1 | 0.00000 | 0.00000 | 0.00000 | ||
Te1 | 0.00000 | 0.00000 | 0.87066 | ||
S1 | 0.00000 | 0.00000 | 0.29134 |
Pb2Bi4Te4S4 | c = 80.73010 Å |
Mr = 1888.98 | V = 1276.06 Å3 |
Trigonal, R3m | Z = 6 |
a = 4.27221 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.00000 | 0.00000 | −0.06385 | ||
Bi2 | 0.00000 | 0.00000 | 0.14517 | ||
Pb1 | 0.00000 | 0.00000 | 0.64551 | ||
Te1 | 0.00000 | 0.00000 | 0.58074 | ||
Te2 | 0.00000 | 0.00000 | 0.78975 | ||
S1 | 0.00000 | 0.00000 | 0.29120 | ||
S2 | 0.00000 | 0.00000 | 0.00000 | ||
S3 | 0.00000 | 0.00000 | 0.50000 |
Pb4Bi4Te4S6 | c = 16.94310 Å |
Mr = 2367.51 | V = 266.02 Å3 |
Trigonal, P3m1 | Z = 1 |
a = 4.25788 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.00000 | 0.00000 | 0.19477 | ||
Pb1 | 0.33333 | 0.66667 | 0.60123 | ||
Te1 | 0.33333 | 0.66667 | −0.08940 | ||
S1 | 0.33333 | 0.66667 | 0.29817 | ||
S2 | 0.00000 | 0.00000 | 0.50000 |
Pb4Bi4Te4S6 | c = 33.81797 Å |
Mr = 2367.51 | V = 531.62 Å3 |
Trigonal, P3m1 | Z = 2 |
a = 4.26052 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.33333 | 0.66667 | 0.20348 | ||
Bi2 | 0.33333 | 0.66667 | 0.39789 | ||
Pb1 | 0.33333 | 0.66667 | 0.89865 | ||
Pb2 | 0.00000 | 0.00000 | 0.00000 | ||
Pb3 | 0.00000 | 0.00000 | 0.50000 | ||
Te1 | 0.33333 | 0.66667 | 0.74380 | ||
Te2 | 0.00000 | 0.00000 | 0.34516 | ||
S1 | 0.00000 | 0.00000 | 0.84818 | ||
S2 | 0.33333 | 0.66667 | 0.05060 | ||
S3 | 0.33333 | 0.66667 | 0.55025 |
Pb6Bi4Te4S8 | c = 61.24237 Å |
Mr = 2846.04 | V = 958.12 Å3 |
Trigonal, R3m | Z = 3 |
a = 4.25029 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.00000 | 0.00000 | 0.05416 | ||
Pb1 | 0.00000 | 0.00000 | 0.77739 | ||
Pb2 | 0.00000 | 0.00000 | 0.50000 | ||
Te1 | 0.00000 | 0.00000 | 0.69161 | ||
S1 | 0.00000 | 0.00000 | 0.41621 | ||
S2 | 0.00000 | 0.00000 | 0.13877 |
Pb8Bi4Te4S10 | c = 71.45943 Å |
Mr = 3324.57 | V = 1115.07 Å3 |
Trigonal, R3m | Z = 3 |
a = 4.24478 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.00000 | 0.00000 | 0.54632 | ||
Pb1 | 0.00000 | 0.00000 | −0.07188 | ||
Pb2 | 0.00000 | 0.00000 | 0.30940 | ||
Te1 | 0.00000 | 0.00000 | 0.85456 | ||
S1 | 0.00000 | 0.00000 | 0.23765 | ||
S2 | 0.00000 | 0.00000 | 0.61879 | ||
S3 | 0.00000 | 0.00000 | 0.00000 |
Pb10Bi4Te4S12 | c = 27.31123 Å |
Mr = 3803.10 | V = 425.78 Å3 |
Trigonal, P3m1 | Z = 1 |
a = 4.24282 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.00000 | 0.00000 | 0.12157 | ||
Pb1 | 0.33333 | 0.66667 | 0.75138 | ||
Pb2 | 0.33333 | 0.66667 | 0.37407 | ||
Pb3 | 0.00000 | 0.00000 | 0.50000 | ||
Te1 | 0.33333 | 0.66667 | −0.05593 | ||
S1 | 0.33333 | 0.66667 | 0.18609 | ||
S2 | 0.00000 | 0.00000 | 0.31150 | ||
S3 | 0.33333 | 0.66667 | 0.56288 |
Pb12Bi4Te4S14 | c = 92.27622 Å |
Mr = 4281.63 | V = 1437.84 Å3 |
Trigonal, R3m | Z = 3 |
a = 4.24175 Å | T = 0 K |
x | y | z | Biso*/Beq | ||
Bi1 | 0.00000 | 0.00000 | 0.03594 | ||
Pb1 | 0.00000 | 0.00000 | 0.74023 | ||
Pb2 | 0.00000 | 0.00000 | 0.44421 | ||
Pb3 | 0.00000 | 0.00000 | 0.14810 | ||
Te1 | 0.00000 | 0.00000 | 0.68316 | ||
S1 | 0.00000 | 0.00000 | 0.38839 | ||
S2 | 0.00000 | 0.00000 | 0.09224 | ||
S3 | 0.00000 | 0.00000 | 0.79617 | ||
S4 | 0.00000 | 0.00000 | 0.50000 |
Acknowledgements
We acknowledge access to the Phoenix high-performance computer at the University of Adelaide. Fabien Voisin and Mark Innes are thanked for their assistance with VASP installation and HPC configuration. We thank two anonymous reviewers for the insightful comments and editorial handling by Michal Dušek.
Funding information
This is a contribution to the Australian Research Council Linkage Project LP200100156 (Critical Metals from Complex Copper Ores) co-supported by BHP Olympic Dam.
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