We have performed a detailed structural analysis of several crystals from the Sb2Te3−xSex solid solution (with x = 0–1.55) using a combination of single-crystal and powder X-ray diffraction and high-resolution scanning transmission electron microscope imaging combined with energy-dispersive X-ray spectroscopic mapping. The experimental study was supported by quantum-chemical calculations. All compounds crystallize in the rhombohedral tetradymite structure type in which the atoms occupy three symmetrically independent crystallographic sites; the two chalcogenide sites A1 and A2, and the antimony site Sb. They form quintuple A2–Sb–A1–Sb–A2 layers perpendicular to the  direction, held together via A2–A2 chalcogenide-chalcogenide interactions. The results of our experimental study show unambiguously, and in agreement with quantum-chemical calculations, a preferred incorporation of selenium into the A2 position inside the layers. An unexpected increase of the van-der-Waals inter-layer distances is observed for low Se contents (x < 1) and discussed with the help of DFT calculations.
Antimony telluride, Sb2Te3, has been extensively studied due to its extraordinary physical properties like thermo-electricity , , as a topological insulator , or for its potential application in non-volatile phase-change memory devices . It has been demonstrated that the phase-change properties of Sb2Te3 can be efficiently tuned by cationic substitution with elements such as Ge, In, or Ag . Substitutions on the chalcogenide sites are less common but may also lead to interesting property changes. The pseudo-binary Sb2Te3−xSex solid solution is a key system in this respect, and members of this system are also discussed as potential phase-change materials  or as topological insulators . Unfortunately, a precise crystallographic characterization of the solid solution was missing up to now, a fact that strongly hampers the establishment of reliable structure-property relationships.
The crystal structure of Sb2Te3 (space group R3̅m, No. 166) was first determined in the 1950s . Within the structure three symmetry-independent sites are occupied: one antimony site (Wyckoff position 6c with z=0.399; site symmetry 3m) and two chalcogenide sites (A1 on 3a, site symmetry 3̅m, and A2 on 6c with z=0.787, site symmetry 3m) occupied by Te. The atoms form quintuple A2–Sb–A1–Sb–A2 layers which are stacked along the c axis and held together via homoatomic, seemingly van-der-Waals-like A2–A2 interactions (Fig. 1a). The coordination polyhedra of each type of atom are shown in Fig. 1b. The Sb atoms are coordinated by three A1 and three A2 atoms resulting in a distorted octahedron. The A1 atoms are surrounded by six Sb atoms in a regular octahedral coordination whereas the A2 atoms are coordinated by three Sb atoms plus three longer bonds to three A2 atoms, beyond the van-der-Waals interaction limit, resulting altogether in a strongly distorted octahedral motif.
The pseudo-binary phase diagram of Sb2Te3−xSex was first studied in the 1960s ,  and exhibits two single-phase regions. Crystals with x of up to 1.8 are isostructural to Sb2Te3 and form a solid solution in the tetradymite or Bi2Te3 structure type. Another single-phase region was found for high x values between 2.85 and 3 representing a solid solution in the orthorhombic Sb2Se3 structure type . In-between these two single-phase regions (1.8<x<2.85) a miscibility gap has been identified, and a two-phase region in which the Sb2Te3 and Sb2Se3 structures co-exist was postulated.
By substituting tellurium by selenium, a preferred incorporation of selenium into the A1 position inside the quintuple layers was postulated by Ullner already in 1968 . In 1974, Anderson et al. refined the crystal structure of Sb2Te2Se based on single-crystal data and claimed a Se/Te order in which Se exclusively occupies the A1 site while tellurium is located at A2 . In that publication, however, one does not find any information as to whether the corresponding occupancies of the two anionic sites were refined or simply assumed by the authors.
In this work, we have performed a detailed structural analysis of several compounds in the stability field of the tetradymite structure type with representative compositions. The experimental investigations were supported by quantum-chemical calculations. Our study focuses on the changes in bonding behavior which are induced by the incorporation of selenium into the Sb2Te3 structure.
2.1 Synthesis, single-crystal and powder diffraction
In a first step, powders of the end members Sb2Te3 and Sb2Se3 were synthesized from stoichiometrically weighed amounts of the elements (Sb: 99.9999%, Koch-Light Laboratories Limited; Se: 99.999%, MaTeck; Te: 99.999% ChemPur) in evacuated quartz ampoules at T=973 K. Polycrystalline samples of Sb2Te3−xSex (x=0–1.8) were synthesized from stoichiometric amounts of the binary phases. For this, pellets of the starting materials were placed in glass ampoules which were sealed under Ar atmosphere. The ampoules were heated up to 773 K for 7 days and subsequently quenched to room temperature in air.
Low-temperature powder-diffraction measurements of the ground samples Sb2Te3, Sb2Te2.4Se0.6, Sb2Te1.8Se1.2, and Sb2Te1.2Se1.8 were performed between 20 and 300 K, using a Huber G670 Guinier Camera equipped with a Ge(111) crystal monochromator, a closed-cycle He cryostat, and an image-plate detector. The samples were cooled down to T=20 K with cooling rates of 5 K·min−1 and then equilibrated for 20 min. Powder patterns (CuKα1 radiation; 0–100° in 2θ, Δθ=0.005°) were measured in steps of 5 K upon heating with a rate of 5 K·min−1 between each temperature point (dwell time of 5 min).
For the low-temperature powder-diffraction measurements, the lattice parameters were determined from Le Bail refinements  at each temperature using the program Jana2006 . The background was described by 100 points and automatically fitted using a 20-parameter polynomial function. A pseudo-Voigt profile function with an angle-independent Lorentzian part and two Gaussian parameters (one angle-dependent and one angle-independent) were used, and the zero shift was refined. Lattice parameters given by Mansour et al.  for Sb2Te3 were used as starting values.
Single crystals of Sb2Te3−xSex were grown by chemical vapor transport from the corresponding polycrystalline samples using iodine (99.5%, Merck) as transport agent at temperatures from 773 to 753 K over 18 h.
Single-crystal diffraction data were measured on a Bruker SMART APEX diffractometer with MoKα radiation. The integration was performed with Saint+, and a multi-scan absorption correction using the program Sadabs ,  was applied. The structure was solved using direct methods (Shelxs ). Subsequent least-squares refinements based on F2 were performed with the program Shelxl . Selected crystallographic data are given in Table 1.
|Lattice parameter a, Å||4.2714(6)||4.2459(3)||4.2132(6)||4.201(3)||4.1919(5)||4.1757(3)||4.1646(3)||4.1556(11)|
|Volume V, Å3||481.3(2)||473.66(10)||462.72(19)||458.7(5)||455.61(10)||450.91(8)||447.50(8)||445.3(2)|
|Density d, g cm−3||6.483||6.401||6.418||6.382||6.321||6.280||6.252||6.164|
|Abs. coeff. μ, mm−1||21.643||22.439||23.294||23.726||24.133||24.644||25.015||24.707|
|Tmin; Tmax||0.4847; 0.7461||0.4303; 0.7461||0.5280; 0.7459||0.4305; 0.7461||0.3808; 0.7461||0.3808; 0.7461||0.2296; 0.7461||0.3872; 0.7460|
|hkl range||−5, 6;||−6, 5;||−3, 5;||−6, 5;||−6, 4;||−5, 2;||−5, 5;||−5, 5;|
|−6, 3;||−4, 5;||−5, 5;||−5, 5;||−3, 5;||−5, 6;||−5, 5;||−5, 5;|
|−39, 42||−33, 42||−42, 40||−41, 30||−41, 41||−40, 41||−40, 34||−41, 29|
|No. of refl.||1304||1266||1234||956||1222||941||1183||1147|
|No of indep. refl.||227||227||212||216||215||209||213||209|
|No of parameters||10||11||11||11||12||11||12||12|
|R1; wR2 (all data)||0.0202; 0.0426||0.0243; 0.0513||0.0433; 0.0692||0.0402; 0.0762||0.0165; 0.0358||0.0259; 0.0581||0.0271; 0.0645||0.0299; 0.0675|
|Occupancies of Te|
|Anisotropic displacement parameters|
Further details concerning data collection and refinement can be obtained from the Fachinformationszentrum Karlsruhe, 76344 Eggenstein-Leopoldshafen, Germany (fax: +49-7247-808-666; e-mail: email@example.com) on quoting the deposition numbers CSD-1898610–1898617.
To independently validate the XRD results in real space, phases with the compositions Sb2Te2.4Se0.6, Sb2Te1.8Se1.2 and Sb2Te1.2Se1.8 were studied by TEM. For TEM measurements lamellae were prepared using a focused ion beam (FIB) technique employing a FEI Dual Beam Helios NanoLab system. The lamellas were cut along the c axis of Sb2Te3−xSex single crystals.
The High Resolution High Angle Annular Dark Field Scanning Transmission Electron Microscopy (HR HAADF STEM) combined with STEM energy-dispersive X-ray spectroscopy (STEM-EDX) mappings was carried out on an FEI Titan ChemiSTEM probe Cs-corrected TEM operated at 200 kV . For EDX mapping a Bruker “Super-X” wide solid angle EDX detector was used at a voltage of 200 kV .
2.3 Quantum-chemical calculations
Density-functional theoretical calculations were carried out using the Vienna ab initio simulation package (Vasp) . The projector-augmented wave method  was used together with the local-density approximation  to account for exchange and correlation. Plane-wave basis sets with kinetic energies up to 500 eV were used.
The structures for the solid-solution phases were set up based on a 2×2×1 supercell of Sb2Te3 in the hexagonal setting containing 12 formula units and 60 atoms. For each composition, six setups were created, three models with a random substitution and three with partially ordered substitution of Se for Te. For the “disordered” model, the Te atoms were replaced randomly by Se with increasing Se content x. For the partially “ordered” structures, first the tellurium atoms on the Te1 site were replaced by Se and, once the site was fully occupied by Se (x=1), the tellurium atoms on the Te2 sites were randomly substituted by selenium.
3 Results and discussion
Figure 2a shows the Se site-occupancy factors of the A1 and A2 sites obtained from our single-crystal structure refinements as a function of the selenium content x. Clearly, for x≤1 the selenium atoms are preferably incorporated into the A1 position, however, a refinement of the site occupancy factor of the A2 site indicates that there is also a small selenium content (below 5%) on this site for all investigated crystals, which was not reported before. For higher Se contents (x>1), the Te on the A2 position is substituted by Se. It is noteworthy that even for the highest selenium contents, a small percentage of tellurium is still present on the A1 position. Our single-crystal data of the Sb2Te3−xSex samples with x=0–1.55 therefore clearly shows a preferred incorporation of selenium into the A1 position and, thus, they are in full accordance with the literature .
Independent theoretical electronic-structure considerations confirm the preferred incorporation of selenium at the A1 position. In Fig. 2b we present the relative formation energies ΔE of Sb2Te3−xSex solid-solution crystals with numerical entries for randomly distributed and partially ordered Se/Te occupation. These energies are related to those of the end member phases, Sb2Te3 and Sb2Se3, where the latter was also referred to the rhombohedral Bi2Te3 structure type, via ΔE=E(Sb2Te3−xSex)−(3−x)/3 E(Sb2Te3)−x/3 E(Sb2Se3). The data clearly reflect that there is a gain in energy by the preferred incorporation of selenium into the center of the quintuple layers, the minimum of ΔE=8.3 kJ mol−1 occurring at x=1. For this composition, the (theoretical) structure is completely ordered with selenium at the A1 site and tellurium on the A2 site. Obviously, a random Se incorporation leads to energies which are both higher than the energies of (a) the pure end members and of (b) the solid-solution phases assuming an ordered distribution of the two chalcogenide anions.
Figure 3 shows HR HAADF STEM Z-contrast images of pristine Sb2Te3 and Sb2Te1.2Se1.8 taken along the  axis together with the corresponding integrated line profiles along the c axis. The image reflects the tetradymite structure of the Sb2Te1.2Se1.8 sample. The intensities seen in these HAADF STEM images are roughly proportional to the atomic number Z1.7 ; hence they enable a direct interpretation. The lighter Sb has a slightly smaller atomic number than Te (ZSb=51 and ZTe=52) and, therefore, it appears as darker spots when compared to tellurium. Nonetheless, the A1 and A2 positions even in a pristine Sb2Te3 specimen appear with a different contrast due to the different coordination. A1 placed in the middle of the quintuple block shows a higher contrast than the A2 position which is situated next to the van-der-Waals gap.
In contrast to the end member Sb2Te3, the Sb2Te1.2Se1.8 sample shows the opposite tendency because the Sb sites appear brighter than the Te sites. This already confirms that the lighter Se (ZSe=34) is incorporated into the Te positions. Besides that, the line scan shows a lower intensity for the A1 position when compared to A2, indicating the preferred incorporation of selenium at A1.
Figure 4a shows the HAADF STEM Z-contrast image of the mapped region together with EDX of elemental Sb, Te, Se and composite HAADF STEM/Sb/Te/Se maps of the Sb2Te1.8Se1.2 sample. The Se-filtered map and the composite figure clearly visualize that the A1 position inside the layer is mainly occupied by selenium, while the outer A2 position is characterized by a mixed occupancy of selenium and tellurium. In other words, these results are fully matching those from the single-crystal XRD measurements. The almost perfect match between the XRD data and the EDX mapping continues for the Sb2Te2.4Se0.6 and Sb2Te1.2Se1.8 compositions, as seen from Fig. 4b.
The interatomic distances and bond angles of several Sb2Te3−xSex crystals are presented in Fig. 5a–c. Because of the high symmetry of the tetradymite structure, only two symmetry-independent Sb–A distances (dSb−A1; dSb−A2) and three interatomic angles (A1–Sb–A1; A2–Sb–A2 and A1–Sb–A2) around the Sb position are to be considered (Fig. 5a). Figure 5b shows the heteroatomic distances dSb−A1 and dSb−A2 as a function of the compositional parameter x. For pure Sb2Te3 the distance dSb−A2=2.9954(5) Å is close to the ideal distance between ions of the two elements expected for an octahedrally coordinated Sb–Te system (2.97 Å) , while dSb−A1 is significantly longer (3.17 Å), indicating a different bonding nature. For small selenium contents, the distance between Sb and A1 decreases upon increasing Se content, while dSb−A2 seems to be nearly constant up to x=1. Obviously, the preferred selenium incorporation inside the layers affects only the Sb–A1 bond whereas dSb−A2 remains essentially unchanged. Above x=1 the A1 position is almost completely occupied with selenium and the Sb–A1 distance remains constant and, thus, the bond length is not affected by the Se incorporation into the A2 site. On the other hand, dSb−A2 at the periphery of the layers starts to slightly decrease with an increasing Se content on this position.
For the solid-solution phases, the values for dSb−A1 and dSb−A2 can also be derived from DFT. The calculations arrive at two values for each Sb–A distance, one for the Sb–Se and one for the Sb–Te combination. As depicted in Fig. 5b, for the end member Sb2Te3, the dSb−Te2 distance is in perfect agreement with the experimental data, while the distance dSb−Te1 is slightly underestimated. The theoretically predicted value for the dSb−Te2 distance is in good agreement with experimental data up to a Se content of x<1 (where hardly any Se is incorporated into the A2 site). For higher Se contents (x>1), when Se starts to occupy the A2 site, the theoretical dSb−Te distance is slightly overestimated, while at the same time the theoretical dSb−Se2 distance is clearly too small. As the diffraction experiment only sees an Se/Te average position with Te being the stronger scatterer, the smaller Sb–Se2 distance as given by theory is less visible within XRD. In the same spirit, the theoretically predicted Sb–Se1 distance for low Se contents (x<1) is smaller than the average value, while the theoretical Sb–Te1 distance is closer to the experiment. Nonetheless, the general trend of the Sb–A1 distance as a function of Se content is well reproduced throughout.
In Fig. 5c, the experimentally determined angles A1–Sb–A1, A1–Sb–A2 and A2–Sb–A2 are shown as a function of the selenium content. For pure Sb2Te3 we observe a maximum deviation between the three angles indicating the largest degree of octahedral distortion. By replacing tellurium in the A1 position with selenium, the distortion of the octahedra decreases, and the angles are getting closer to the ideal octahedral angle of 90°. For the almost completely ordered compound Sb2Te2Se, we find the minimum angular spread and the smallest deviations from 90°. For Se contents x>1, the distortion of the octahedra starts to increase again.
Figure 6a shows the experimental and theoretically predicted interlayer distances dA2−A2 as a function of the selenium content. The experimental data exhibit an unexpected behavior while the selenium amount increases. In contrast to the heteroatomic distances, dA2−A2increases with increasing selenium content up to a maximum value of x=1. This is rather unexpected since – as we have shown before – for these selenium contents, there is next to no Se present on the A2 site and thus, one would expect that the interlayer distances remain unchanged. This strange behavior indicates, however, that an incorporation of selenium into the central position of the layer weakens the interaction between the layers. The theoretical data fully corroborate the experimental observation, and also the interatomic distances reported in the literature for Sb2Te3 and Sb2Te2Se are in good agreement  (see Fig. 6a).
Recently, low-temperature heat-capacity data of several Sb2Te3−xSex solid-solution crystals have been reported  and it was concluded that the incorporation of selenium into the Sb2Te3 structure type leads to an increasing bond polarity as a consequence of the replacement of Te by the more electronegative selenium. The unexpected elongation of the interlayer distance dA2−A2 can be explained, at least qualitatively, as a secondary effect: the increasing polarity inside the layer attracts more electron density from the periphery of the layer and leads to a weakening of the interlayer interactions. At x values larger than 1, the interlayer distance shrinks again because the smaller Se enters the A2 position and, thus, weakens the repulsion between neighboring layers. As pointed out already, the relative change of dA2−A2 is perfectly reproduced in the values obtained from electronic-structure theory, despite the fact that the absolute values are slightly underestimated. This underestimation is due to the fact that the van-der-Waals gaps are hard to model by quantum-chemical calculations based on density-functional theory. We have recently proven the reliability of the LDA functional for the case of Sb2Te3 where the lattice parameters as given by LDA calculations are qualitatively comparable to those using explicit van-der-Waals corrections , so a different (and far more expensive) theoretical approach is unlikely to further improve the agreement between experiment and theory.
Intuitively, one might consider the interlayer distance dA2−A2 in Sb2Te3 as a typical van-der-Waals contact – as was indeed assumed in the literature – but this looks like an oversimplification in the present case. The van-der-Waals radius of tellurium according to Bondi is 2.06 Å  and, therefore, one would expect a Te–Te van-der-Waals distance around 4.12 Å. However, from our structural analysis we have determined a value of 3.7120(7) Å for the A2–A2 bond length which is about 10% smaller than the expected value. In other words, there must be some additional attractive contribution to the weak Te–Te interactions which leads to a pronounced shortening of these bonds.
The ideal van-der-Waals bond length dvdw for the Sb2Te3−xSex solid-solution crystals can be calculated from the site occupancy factor xCh and the Bondi radii of the chalcogenide rCh as an arithmetic average:
The ratio between the experimentally determined A2–A2 distance and the ideal van-der-Waals bond length (dvdW), calculated from the experimentally refined occupancies of the A2 position, is shown in Fig. 6b. For all compounds, the ratio is clearly below 1, so that all experimental A2–A2 distances are “too short” with respect to the corresponding ideal value. The ratio shows an almost linear increase, with a little shoulder at x=1, indicating that incorporating selenium results in a weakening of the interlayer interactions which, once again, must be stronger than what would be expected from a typical van-der-Waals contact. Also, this behavior depends on the selenium content, rather than on the preferred selenium incorporation on the A1 position. As mentioned before, the elongation of the A2–A2 distance can be explained by the increasing polarity of the antimony-chalcogenide bonds inside the layers, but it does not explain why these bonds are much shorter than expected for a van-der-Waals contact in the first place.
In order to understand the origin of the comparably short A2–A2 interactions in detail, we have compared the interlayer distances of several compounds with the tetradymite structure type (and related structures such as Ge2Sb2Te5) to the corresponding value determined for Sb2Te3 (Table 2). The too short interlayer van-der-Waals bond lengths for some of those materials suggest the presence of additional attractive interactions between the layers that go beyond classical van-der-Waals forces. It should be pointed out in this context that the unusual behavior of some of these compounds and the relation to the underlying bonding scheme are currently under discussion  and are discussed to be partly responsible for the interesting physical properties (e.g. thermoelectricity, phase-change applications, topological insulators) which are not observed for materials with classical van-der-Waals bonds.
The experimentally determined c/a lattice-parameter ratio obtained from powder-diffraction measurements are shown in Fig. 7 and compared to the corresponding data from theoretical calculations. Due to the strongly anisotropic nature of the crystal structure, large c/a ratios of about 7 were found. With increasing selenium content, the c/a ratios increase. Around x=1 one observes a slight shoulder in the increase which is also reflected in the theoretical data. The absolute computational values are smaller than the experimental ones, which is due to the fact that the van-der-Waals interactions along c are hard to describe by quantum-chemical calculations.
The increasing c/a parameter as a function of the compositional parameter x indicates that the decrease of a is stronger than the decrease of the c, as also reflected from the individual bond lengths derived from the single-crystal diffraction measurements. The homoatomic A2–A2 van-der-Waals-like bonds mainly contribute to the c lattice parameter, while the heteroatomic bonds are mostly oriented in the ab plane, contributing mainly to the a parameter. Therefore, the observed increase of dA2−A2 for x<1, together with the decrease of the heteroatomic bonds, lead to the observed increase of c/a. The absolute changes of the heteroatomic bond length dSb−A1 and dSb−A2 are much larger than those for dA2−A2 leading to an increase of c/a even for x>1, where both bond length decrease.
Figure 7b shows the temperature-dependent c/a ratios for selected compounds from the Sb2Te3−xSex system. For Sb2Te3, the values are compared to literature values . With decreasing temperature, the c/a ratio shows an almost linear decrease down to around 50 K for all samples. In general, the thermal behaviour demonstrates that the decrease in the c lattice parameter as a function of decreasing temperature is more pronounced than the decrease in the a lattice parameter. This suggests that the van-der-Waals-like interactions between the layers are more influenced by the temperature than the strong heteroatomic covalent bonds within the layer. The change in trend for c/a around T=50 K could indicate that the van-der-Waals-like interactions reach their energetic minimum (strongest interaction) at this temperature.
For the first time, a detailed structural characterization of the solid solution Sb2Te3−xSex with x=0–1.5 adopting the Bi2Te3 structure type was performed using a combination of single-crystal and temperature-dependent X-ray diffraction measurements, HR-TEM imaging, EDS mapping, and quantum-chemical calculations. Both experimental and theoretical results indicate a preferred incorporation of selenium into the central A1 site of the quintuple layers. The A2–A2 interlayer distance in Sb2Te3 is significantly smaller than expected for a conventional Te–Te van-der-Waals interaction. This behavior can also be observed for similar tetradymite-type compounds despite the fact that other layered structure types exhibit expected van-der-Waals distances. An unexpected increase of the interlayer distances in Sb2Te3−xSex was observed upon Se incorporation and leads to a maximum distance for x=1. We explain this phenomenon by the increasing binding polarity of the Sb–A1 and Sb–A2 bonds which pulls electron density into the layers and, therefore, weakens the interlayer bonding.
Dedicated to: Professor Arndt Simon on the occasion of his 80th birthday.
This work was supported by the Deutsche Forschungsgemeinschaft within the SFB Nanoswitches (SFB 917). We thank the Jülich-Aachen Research Alliance (JARA) as well as the RWTH Aachen University IT Center for providing CPU time within the JARA-HPC project “jara0033”.
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