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Crystal and electronic structures and physical properties of two semiconductors: Pb4Sb6Se13 and Pb6Sb6Se17
Intermetallics 14 (2006) 198–207 www.elsevier.com/locate/intermet
Crystal and electronic structures and physical properties of two semiconductors: Pb4Sb6Se13 and Pb6Sb6Se17 Shahab Derakhshan, Abdeljalil Assoud, Nicholas J. Taylor, Holger Kleinke* Department of Chemistry, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received 26 April 2005; accepted 24 May 2005 Available online 28 July 2005
Abstract The title compounds were synthesized by directly reacting the elements in stoichiometric ratios at elevated temperatures. Their crystal structures were determined by single crystal X-ray diffraction. Pb4Sb6Se13 crystallizes in the monoclinic space group I2/m with lattice ˚ , bZ4.0910(2) A ˚ , cZ25.212(1) A ˚ , bZ93.943(1)8, VZ2530.3(2) A ˚ 3 (ZZ4), while Pb6Sb6Se17 crystallizes in dimensions of aZ24.591(1) A ˚ , bZ24.061(7) A ˚ , cZ4.1382(9) A ˚ , VZ1580.4(7) A ˚3 the orthorhombic space group P21212 with lattice dimensions of aZ15.872(4) A (ZZ2). Electronic structure calculations predicted semiconducting behavior. Temperature dependent electrical conductivity measurements verified this prediction for Pb4Sb6Se13. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Ternary alloy systems; B. Electronic structure; B. Electrical resistance; B. Thermoelectric properties
1. Introduction Thermoelectrics are materials that are able to convert thermal energy into electricity and vice versa. They are used for power generation utilizing the Seebeck effect and for heat pumping and refrigeration (Peltier effect). The applicability of these compounds is mostly limited because of their relatively low efficiency. Their efficiency depends on the dimensionless fig.-of-merit, ZT, defined as ZTZ TS2s/k. Therein, T is the absolute temperature, S is the thermopower or the Seebeck coefficient, s is the electrical conductivity, and k is the thermal conductivity [1]. Therefore, high thermopower and high electrical conductivity combined with low thermal conductivity are ideal. Since both electrons and phonons contribute to the thermal conductivity, lowering the electrical contribution, ke, will reduce and thus negatively affect s as well. The challenge for materials chemists in this field remains in the optimization of these transport properties by synthesizing materials with high S, high s and low kph. Hence, promising
* Corresponding author. E-mail address: [email protected] (H. Kleinke).
0966-9795/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2005.05.008
materials are narrow band gap semiconductors composed of heavy elements [2–4]. Binary lead, bismuth and antimony chalcogenides have been commercially used for thermoelectric applications for a long time. Moreover, the most studied materials in thermoelectric research include ternary and higher antimony chalcogenides [5–14]. In our research group we have thus far focused on potential thermoelectric materials in the metal tetrelide, pnictide and chalcogenide systems using both experimental and theoretical approaches [15–19]. Currently, we are investigating ternary and higher lead antimony chalcogenides. We recently reported on the thermoelectric properties of FePb4Sb6S14 [20]. Pb4Sb6S13 [21] is a related ternary compound (whose properties were not reported). One can assign the oxidation states of lead, antimony and sulfur to be C2, C3 and K2, respectively. A band gap may be expected to appear between fully occupied S p-states and empty Pb and Sb p-states. Replacing sulfur atoms by selenium atoms would likely shrink the band gap, in favor of higher s by elevating the occupied p-states. At the same time, the ZT value benefits from lowered phonon contribution to the thermal conductivity caused by the heavier chalcogen atom. Here we will introduce the synthesis, crystal structure, electronic band structure, as well as temperature dependent electrical conductivity of the new ternary selenide Pb4Sb6Se13. Furthermore, after we
S. Derakhshan et al. / Intermetallics 14 (2006) 198–207
started our investigation in the Pb/Sb/Se system, the structures of Pb6Sb6S17 [22] and Pb6Sb6Se17 [23] were published almost simultaneously by different authors. According to these reports, they form different structures, the selenide exhibiting Se disorders within a Se2K 3 group. We therefore included Pb6Sb6Se17 in our studies, also because its properties were not reported.
2. Experimental section 2.1. Synthesis All elements were used as purchased from ALFA AESAR with purities between 99.5 and 99.9%. Single crystals of Pb4Sb6Se13 and Pb6Sb6Se17 were obtained by heating stoichiometric mixtures of the elements (overall sample mass per reaction: about 0.5 g) in evacuated quartz tubes to 600 and 800 8C, respectively. At these temperatures the products were molten, and subsequent slow cooling (2 8C/h) yielded needle like crystals. Another sample of Pb4Sb6Se13 was prepared at 600 8C with subsequent faster cooling down to room temperature by switching off the furnace. Phase pure Pb4Sb6Se13 for physical property measurements was synthesized by placing a stoichiometric mixture of Pb, Sb and Se into an alumina crucible, which was then sealed in an evacuated quartz tube and annealed at 500 8C. According to our experiments, Pb6Sb6Se17 melts incongruently, which makes it difficult to prepare pure samples. 2.2. Chemical analysis All samples were investigated by X-ray powder diffraction to check for purity by using an INEL powder diffractometer with a position sensitive detector. EDS analyses (LEO 1530, with integrated EDAX Pegasus 1200) on six selected crystals of Pb4Sb6Se13 revealed the presence of lead, antimony, and sulfur in the ratio of 17:23: 60 (in atomic percent), which compares well with the anticipated ratio of 17:26:57. 2.3. Single crystal structure studies Two black needle-shaped crystals from the Pb4Sb6Se13 samples heated to 600 8C were mounted for a room temperature data collection on a Smart Apex CCD (BRUKER), which utilizes graphite-monochromatized Mo Ka radiation. One crystal was obtained from slow cooling and the other one from faster cooling. In both cases, the crystal-to-detector distance was 4.514 cm. Data were collected by scans of 0.38 in u, for two blocks of 606 frames at fZ0 and 608. The exposure time was 60 s per frame for both crystals. A black needle-shaped crystal from the Pb6Sb6Se17 sample heated to 800 8C was mounted for a room
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temperature data collection on the same diffractometer. Data were collected by scans of 0.38 in u, for an overall of 606 frames at fZ08. The exposure time was 90 s per frame. All three data sets were corrected for Lorentz and polarization effects. Absorption corrections were based on fitting a function to the empirical transmission surface as sampled by multiple equivalent measurements using SADABS [24]. (The numerical absorption correction based on face indexing did not result in any improvement, most likely because the faces could not be determined reliably.) Diffraction peaks obtained from all frames of the reciprocal space images were used to determine the unit cell parameters by a least square analysis. For the structure refinements we employed the SHELXTL [25] program package. For Pb4Sb6Se13 the assumption of monoclinic symmetry, based on the lattice parameters, as well as the systematic absences pointed towards Pb4Sb6Se13 being isostructural to Pb4Sb6S13. The atomic positions were chosen based on the sulfide’s positions. We labeled M1–M4 as in the sulfide, and named the original M10–M15 sites M5–M10 in the selenide. Se1–Se6 correspond to S1–S6, and Se7–Se13 to S10–S16. These sites were arranged in the standard asymmetric unit. We allowed for mixed Pb/Sb occupancies on all M sites, but kept each site overall fully occupied. The decrease of the R value from 0.112 in the completely ordered model down to 0.050 in the model with mixed occupancies proves the validity of our proposed model. In some cases, the Pb/Sb mixing is at the borderline of being significant. For example, the M8 position of the slowly cooled crystal contains only 1.3(6)% Sb, which is about twice the value of its standard deviation (2s). However, the Sb content in this site in the fast cooled crystal reaches a significant value of 7.5(9)%. We list all refined occupancies for better comparison, regardless of their significance. Compared to our model for the selenide, most of the refined positions of the sulfide exhibit higher Pb contents, with the extreme being M15 with 88% Pb, corresponding to our M10 with 54% Pb. Our final refined formula for the slowly cooled crystal is Pb3.98(6)Sb6.02Se13, which is equal to 4: 6: 13 within the standard deviations. The same procedure led to Pb4.02(9)Sb5.98Se13 for the faster cooled crystal, i.e. the same formula within experimental error. On the other hand, the sulfide’s refined formula, Pb5.05Sb4.95S13, is too Pb-rich compared to the expected charge balanced formula. In case of Pb6Sb6Se17, the assumed orthorhombic symmetry was supported by the satisfying internal R value of 0.033 (based on F2). The systematic absences are in accord with the space group P21212 of the analogous sulfide. The refined Flack parameter [26] of 0.02(2) indicated that the absolute structure was correctly determined, and that the structure is not centrosymmetric, as reported for Pb6Sb6Se17 prepared via hydrothermal reaction. Again, we assigned the atoms based on the model published for the sulfide. The M atom positions were examined for mixed occupancy, and the decrease of the R value from 0.068 in the completely
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Table 1 Crystallographic data for Pb4Sb6Se13 and Pb6Sb6Se17 Empirical formula Formula weight (g/mol) Temperature (K) ˚) Wavelength (A Space group ˚) Cell dimensions, a (A ˚) b (A ˚) c (A b (8) ˚ 3) V (A No. of formula units per cell Calculated density (g/cm3) Absorp. coeff. (mmK1) F(000) Crystal size (mm) 2 Theta range Reflections collected Independent reflections (Rint) Obs. data (2s), parameter Absorption correction Min/max. transmission ratio Goodness-of-fit on F2 R(F), Rw(F2) (IO2s(I)) R(F), Rw(F2) (all data) Extinction coefficient ˚ 3) Max. diff. peak, hole (e/A
Pb3.98(6)Sb6.02Se13 2584.03 295(2) 0.71073 I2/m 24.591(1) 4.0910(2) 25.212(1) 93.943(1) 2530.3(2) 4 6.783 51.383 4302 60!3!1 3.24–70.02 13655 5925 (0.041) 4983, 150 Empirical 0.48 1.43 0.0504, 0.1008 0.0604, 0.1039 0.00036(2) 5.13, K4.52
ordered model down to 0.064 supported this assumption in general. The M2, M3, M5 and M6 sites exhibit significant Pb/Sb mixing, while the Pb content of the M4 site of 3.9(6)% may be considered insignificant. Since refining the
Pb6.12(6)Sb5.88Se17 3326.64 298(2) 0.71073 P21212 15.872(4) 24.061(7) 4.1382(9) 1580.4(7) 2 6.991 56.998 2760 70!3!2 3.08–69.98 8613 5221 (0.033) 4220, 138 Empirical 0.74 1.25 0.0643, 0.0931 0.0780, 0.0968 0.00027(2) 3.94, K4.23
occupancy factor of M1 yielded 100.7(7)% Pb, we fixed it as a Pb site, i.e. Pb1. The so-refined formula, Pb6.12(6)Sb5.88Se17, is within 2s of the charge balanced 6:6:17 ratio. Crystallographic details of these measurements are
Table 2 Atomic coordinatesa and equivalent displacement parameters of Pb4Sb6Se13 obtained via slow cooling, and occupancy factors of both cases as well as of Pb4Sb6S13 Atom
x
z
˚ 2) Ueq (A
Slow cooling occ. Pb
Fast cooling occ. Pb
Pb4Sb6S13 occ. Pb
Atom
M(1) M(2) M(3) M(4) M(5) M(6) M(7) M(8) M(9) M(10) Se(1) Se(2) Se(3) Se(4) Se(5) Se(6) Se(7) Se(8) Se(9) Se(10) Se(11) Se(12) Se(13)
0.1925(1) 0.0065(1) 0.4371(1) 0.3823(1) 0.0107(1) 0.0798(1) 0.1581(1) 0.1791(1) 0.2730(1) 0.3618(1) 0.1938(1) 0.0265(1) 0.4188(1) 0.4045(1) 0.2339(1) 0.1383(1) 0.0897(1) 0.1724(1) 0.2501(1) 0.3020(1) 0.3799(1) 0.4577(1) 0.4627(1)
0.2997(1) 0.3137(1) 0.9448(1) 0.0775(1) 0.8524(1) 0.9889(1) 0.8278(1) 0.6603(1) 0.5223(1) 0.3692(1) 0.4109(1) 0.4278(1) 0.8430(1) 0.1793(1) 0.1853(1) 0.5466(1) 0.1927(1) 0.0452(1) 0.8911(1) 0.7298(1) 0.5784(1) 0.4357(1) 0.7124(1)
0.032(1) 0.027(1) 0.029(1) 0.033(1) 0.030(1) 0.035(1) 0.028(1) 0.029(1) 0.024(1) 0.032(1) 0.018(1) 0.018(1) 0.017(1) 0.017(1) 0.021(1) 0.017(1) 0.021(1) 0.017(1) 0.016(1) 0.022(1) 0.019(1) 0.020(1) 0.028(1)
0.263(5) 0.787(6) 0.015(6) 0.036(6) 0.176(6) 0.076(6) 0.194(6) 0.987(6) 0.904(6) 0.542(6)
0.300(8) 0.721(9) 0.043(8) 0.067(9) 0.152(8) 0.172(9) 0.194(8) 0.925(9) 0.867(9) 0.582(9)
0.41 0.85 0.08 0.16 0.16 0.24 0.27 1.00 1.00 0.88
M(1) M(2) M(3) M(4) M(10) M(11) M(12) M(13) M(14) M(15) S(1) S(2) S(3) S(4) S(5) S(6) S(10) S(11) S(12) S(13) S(14) S(15) S(16)
a
All atoms on 4i, hence yZ0.
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Table 3 Atomic coordinates and equivalent displacement parameters of Pb6Sb6Se17 Atom
Wyck.
x
y
z
˚ 2) Ueq (A
Occ. Pb
Pb(1) M(2) M(3) M(4) M(5) M(6) Se(1) Se(2) Se(3) Se(4) Se(5) Se(6) Se(7) Se(8) Se(9)
4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 2a
0.42210(4) 0.31945(4) 0.29823(4) 0.17250(7) 0.43883(7) 0.56345(6) 0.40671(8) 0.22781(9) 0.28859(8) 0.31492(8) 0.55067(8) 0.12016(8) 0.48651(9) 0.60512(9) 1/2
0.37208(2) 0.20317(2) 0.53522(2) 0.39546(5) 0.07004(5) 0.24314(4) 0.27801(5) 0.32950(5) 0.11378(5) 0.44606(5) 0.42535(5) 0.49148(6) 0.15130(5) 0.31407(6) 1/2
0.5050(2) 0.4950(3) 0.5111(3) 0.0339(4) 0.9947(6) 0.0054(5) 0.9993(6) 0.5216(6) 0.9868(6) 0.0200(5) 0.0177(6) 0.5141(6) 0.4812(6) 0.4738(7) 0.6868(5)
0.0296(2) 0.0253(2) 0.0262(2) 0.0340(4) 0.0507(5) 0.0364(4) 0.0162(3) 0.0213(3) 0.0172(3) 0.0153(3) 0.0168(3) 0.0224(3) 0.0210(3) 0.0233(4) 0.0149(5)
1 0.926(6) 0.901(6) 0.039(6) 0.098(7) 0.098(6)
summarized in Table 1. The atomic coordinates, occupancy factors as well as thermal displacement values may be found in Tables 2 and 3. 2.4. Electronic structure calculations We carried out self-consistent tight-binding LMTO calculations (LMTO: linear muffin tin orbitals) [27,28]. In this method, the density-functional theory is applied with the local density approximation (LDA) [29]. The required crystallographic data set for Pb4Sb6S13 was obtained from Skowron et al. The 10 M sites of both Pb4Sb6S13 and Pb4Sb6Se13 were chosen as Sb1, Pb2, Sb3, Sb4, Sb5, Sb6, Sb7, Pb8, Pb9, and Pb10. In this model, Sb rich sites were treated as pure Sb and the rest as pure Pb sites. The same
Fig. 1. Crystal structure of Pb4Sb6Se13 in projections along the b axis. Horizontal: a axis. Small, white circles: Se; gray: Pb/Sb. The larger the Pb content, the larger and darker the circle.
argument was applied for Pb6Sb6Se17, where we assigned Pb1, Pb2, Pb3, Sb4, Sb5, and Sb6 to the respective M sites. Thereby our models have the charge balanced elemental ratios (4:6:13 and 6:6:17). The integration in k space was Table 4 ˚ ) of Pb4Sb6Se13 (slow cooling) Selected bond distances (A M(1)–Se(5) M(1)–Se(1) M(1)–Se(5) M(1)–Se(4) M(2)–Se(2) M(2)–Se(4) M(2)–Se(3) M(2)–Se(13) M(3)–Se(3) M(3)–Se(6) M(3)–Se(2) M(4)–Se(4) M(4)–Se(1) M(4)–Se(2) M(5)–Se(7) M(5)–Se(13) M(5)–Se(12) M(6)–Se(8) M(6)–Se(11) M(6)–Se(12) M(7)–Se(9) M(7)–Se(10) M(7)–Se(11) M(8)–Se(6) M(8)–Se(9) M(8)–Se(3) M(8)–Se(10) M(8)–Se(10) M(9)–Se(11) M(9)–Se(8) M(9)–Se(9) M(9)–Se(1) M(9)–Se(6) M(10)–Se(12) M(10)–Se(7) M(10)–Se(8) M(10)–Se(5)
2x
2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x
2x 2x 2x
2.739(1) 2.804(1) 3.124(1) 3.214(1) 2.884(1) 2.993(1) 3.097(1) 3.339(1) 2.574(1) 2.778(1) 3.055(1) 2.586(1) 2.801(1) 3.0448(1) 2.646(1) 2.725(1) 3.267(1) 2.600(1) 2.877(1) 2.982(1) 2.678(1) 2.729(1) 3.309(1) 2.972(1) 3.034(1) 3.157(1) 3.387(1) 3.450(1) 2.898(1) 3.032(1) 3.073(1) 3.306(2) 3.411(8) 2.799(1) 2.883(1) 3.129(1) 3.341(1)
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˚ ); thin solid: intermediate bonds (2.88–2.98 A ˚ ); thin dashed: long bonds (3.26–3.31 A ˚ ). Fig. 2. Sb–Se substructure of Pb4Sb6Se13. Thick solid: short bonds (!2.80 A
performed by an improved tetrahedron method [30] on grids of 1098, 1098, 228 and 228 independent k points of the first Brillouin zone for Pb4Sb6S13, Pb4Sb6Se13, Pb6Sb6S17 and Pb6Sb6Se17, respectively. The first Brillouin zone (of the primitive reciprocal cell) contains two formula units in each case. 2.5. Physical property measurements We pressed part of the ground phase-pure sample of Pb4Sb6Se13 into a bar-shaped pellet of the dimensions 6!1!1 mm for the temperature dependent electrical conductivity measurements using a four-point-method. A home-made device was used to measure the voltage drops DV over a distance of 2 mm under dynamic vacuum between 275 and 325 K, wherein cooling was achieved by helium compression. Silver paint (TED PELLA) was used to create the electrical contacts.
3. Results and discussion 3.1. Crystal structure of Pb4Sb6Se13 Pb4Sb6Se13 is isostructural with Pb4Sb6S13. The structure (Fig. 1) is composed of ribbons of edge-sharing distorted square-pyramidal MSe5 groups. The M5, M6 and M7 atoms are located in a ribbon with a width of three edge-sharing square-pyramids. The M3 and M4 atoms form another ribbon with a width of two edge-sharing square-pyramids. Each of these five M sites possesses over 80% Sb content. The other M–Se chains interconnect these one-dimensionally extended motifs. Pb atoms dominantly occupy the latter sites. The coordination numbers of the M sites as well as the M–Se distances are directly related to the Pb content in the
positions (Table 4). The sites with Pb content over 50% have coordination numbers of seven and eight. The environment of the M3 and M4 atoms, almost exclusively occupied by Sb atoms, is comparable to the Sb atom within the ribbons of Sb2Se3 [31]. In this binary compound, one of ˚ to the Se atom the Sb atoms forms one short bond of 2.59 A ˚) at the apex of the pyramid, and two intermediate (2.80 A ˚ ) to the Se atoms within the and two long bonds (3.01 A plane of pyramid, i.e. overall a coordination number of 5. Neglecting the two longer bonds, one can describe the structure based on corner-sharing SbSe3 trigonal pyramids, which are interconnected along the b direction. The corresponding trigonal pyramids of Pb4Sb6Se13 are depicted in Fig. 2 by the thick solid lines. The M8 site, occupied by 99% Pb in the slowly cooled crystal, is in an eight-coordinated environment with M–Se ˚ . Such a coordination distances between 2.97 and 3.45 A sphere for Pb is present, for instance, in the crystal structure of PbU2Se5 where eight Pb–Se bond distances are in the ˚ [32]. The M10 atom exhibits range between 2.95 and 3.50 A considerable mixing with 54% Pb and 46% Sb. There are seven M10–Se bonds per M10 atom with distances between ˚ , which are comparable to the M3 atom in 2.80 and 3.34 A PbSb2Se4 [33] (58% Pb content) with seven M3–Se bonds ˚ ). (2.77–3.39 A Since the Pb/Sb ratio is 4:6, statistics dictates that 60% of each of the 10 cation sites will be occupied by Sb. This would be expected, were the entropy the dominating factor. The sites with less than 60% Sb atoms occupancy in the slowly cooled crystal have higher Sb contents in the faster cooled crystal. On the other hand, the sites with more than 60% Sb occupancy in the slowly cooled crystal exhibit lower Sb contribution in the faster cooled crystal. Hence faster cooling yielded a more statistical distribution, i.e. partial ordering occurs during the cooling process. Using the LMTO method we calculated the muffin tin radii for these 10 cation positions. To avoid
S. Derakhshan et al. / Intermetallics 14 (2006) 198–207
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Table 5 ˚ ) vs. Sb occupancies of the M sites of Pb4Sb6Se13 The muffin tin radii (A Site
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
˚) Radius (A Sb slow cooling (%) Sb fast cooling (%)
1.656 0.737(5) 0.700(8)
1.746 0.213(6) 0.279(9)
1.553 0.985(6) 0.937(8)
1.561 0.964(6) 0.933(9)
1.598 0.824(6) 0.848(8)
1.568 0.924(6) 0.828(9)
1.617 0.806(6) 0.806(6)
1.841 0.013(6) 0.075(9)
1.761 0.096(6) 0.133(9)
1.692 0.458(6) 0.418(9)
the effect of the atomic size on the site radius, we assumed all the sites are fully occupied by the Sb atoms. The results are summarized in Table 5. Increasing size corresponds to decreasing Sb contents, as expected based on the smaller size of the Sb atom, compared to Pb.
distances between the two kinds of sites, the Pb/Sb distribution is much more ordered in Pb6Sb6Se17 than in Pb4Sb6Se13, as each site is clearly dominated (by 90% or more) either by Pb or Sb atoms. 3.3. Electronic structures
3.2. Crystal structure of Pb6Sb6Se17 The structure of Pb6Sb6Se17 was originally reported [23] in the space group Pbam. In this model there is a ˚ between two Se9 problematic short contact of 1.54 A atoms. Therefore, the occupancy of Se9 was fixed at 0.5 assuming local ordering that avoids this physically impossible Se–Se distance. Our crystal structure investigation revealed that this compound is isostructural with Pb6Sb6S17, where a model in the space group P21212 with complete S ordering was applied. Hence, our Pb6Sb6Se17 crystallizes in an ordered variant of the hydrothermally prepared Pb6Sb6Se17. The Wyckoff notation of all the atoms is 4c except for the Se9 (2a). Moreover, in contrast to the previously reported model, we noticed that except for Pb1 all the cation sites are mixed occupied by Pb and Sb, with up to 10% Sb on Pb and vice versa (Table 3). The mixing in the M4 site is about 6s and may be insignificant. The structure of Pb6Sb6Se17 (Fig. 3) is composed of ribbons of the width of three edge-sharing distorted squarepyramidal MSe5 groups, which run along the shortest crystallographic axis, c, and Pb rich sites connect these ribbons. An additional feature in the structure of this compound is the presence of the Se2K 3 unit formed by one Se9 atom and two Se5 atoms. Hence the formula may be written as ðPb2CÞ6 ðSb3CÞ6 ðSe2KÞ14 ðSe3 Þ2K. This selenide is structurally related to Sr6Sb6S17 [34]. The unit cell in the latter compound is doubled along the short axis compared to Pb6Sb6Se17, and the space group is P212121. The difference between these two stems from the Sb–Q substructures: the Sb atoms of Pb6Sb6Se17 are aligned in the same line along the shortest axis c (Fig. 4) because of translational symmetry, while the Sb atoms of the Sr compounds form a zigzag chain, thereby causing the cell doubling. Five Se atoms form the coordination spheres of the ˚, positions with high Sb contents (Table 6: 2.57–3.26 A ˚ as a cutoff). Seven and eight Se atoms surround using 3.5 A ˚ ). This correlates nicely to the Pb rich positions (2.93–3.43 A the well-known tendency of the Pb atoms to prefer sites with larger distances and higher coordination numbers. Most likely because of larger differences in the interatomic
The densities of states (DOS) and the band structures of Pb4Sb6S13 and Pb4Sb6Se13, both shown in Fig. 5, are indicative of semiconducting behavior. The Fermi level is located in a gap between fully occupied states of chalcogen Table 6 ˚ ) of Pb6Sb6Se17 Selected bond distances (A Pb(1)–Se(1) Pb(1)–Se(1) Pb(1)–Se(5) Pb(1)–Se(4) Pb(1)–Se(5) Pb(1)–Se(8) Pb(1)–Se(2) Pb(1)–Se(4) Pb(1)–Se(9) M(2)–Se(7) M(2)–Se(3) M(2)–Se(3) M(2)–Se(1) M(2)–Se(1) M(2)–Se(9) M(2)–Se(8) M(3)–Se(4) M(3)–Se(6) M(3)–Se(4) M(3)–Se(3) M(3)–Se(3) M(3)–Se(5) M(3)–Se(5) M(3)–Se(9) M(4)–Se(4) M(4)–Se(2) M(4)–Se(2) M(4)–Se(6) M(4)–Se(6) M(5)–Se(3) M(5)–Se(7) M(5)–Se(6) M(5)–Se(6) M(5)–Se(7) M(6)–Se(1) M(6)–Se(8) M(6)–Se(8) M(6)–Se(7) M(6)–Se(7) Se(5)–Se(9)
2x
3.060(2) 3.092(2) 3.142(2) 3.177(2) 3.211(2) 3.225(2) 3.250(2) 3.256(2) 3.401(1) 2.931(2) 3.001(2) 3.048(2) 3.061(2) 3.085(2) 3.371(2) 3.430(2) 2.967(2) 3.016(2) 3.016(2) 3.117(2) 3.129(2) 3.289(2) 3.325(2) 3.392(2) 2.568(2) 2.714(3) 2.790(3) 3.159(2) 3.264(2) 2.607(2) 2.906(3) 2.929(3) 2.980(3) 2.985(3) 2.626(2) 2.666(3) 2.862(3) 3.202(3) 3.229(2) 2.398(2)
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Fig. 3. Crystal structure of Pb6Sb6Se17 in a projection along the c axis. Horizontal: b axis. Small, white circles: Se; gray: Pb/Sb. The larger the Pb content, the larger and darker the circle.
p-states and empty Sb and Pb p-states. The band gaps in both compounds are indirect. The computed gap size in Pb4Sb6S13 is about 0.8 eV, and in the selenide 0.55 eV. The smallest direct gap size are about 0.9 eV for the sulfide and 0.65 eV for the selenide. The anticipated decrease of the gap size is caused by the valence p-states of the selenide being located at higher energies. The densities of states (DOS) and the electronic band structure of Pb6Sb6S17 and Pb6Sb6Se17 are shown in Fig. 6. Semiconducting properties are to be expected here as well. Indirect band gaps of 1.05 and 0.7 eV and direct band gaps of 1.2 and 0.8 eV are observed for the sulfide and selenide, respectively. The partial DOS calculations prove that the Q atoms dominate the occupied states (in addition to the Sb s-states). It is assumed that the Pb4Sb6Q13 family exhibits smaller gaps than Pb6Sb6Q17 because of the smaller Pb/Sb ratio (due to the lower electronegativity of Pb). We analyzed the Crystal Orbital Hamilton Populations (COHP curves) [35] of selected interactions involving Q5 and Q9 of Pb6Sb6S17 and Pb6Sb6Se17 to investigate the effect of the Q–Q bonds on the band gap size. Since the antibonding states of the Q–Q interactions overlap with the empty Sb p-states, hence occur at the same energies, their impact on the gap size cannot be huge.
ðsZ s0 expðKDE=kTÞÞ, with kZBoltzmann constant and DEZactivation energy [36]. Plotting the electrical conductivity logarithmically as a function of 1/T (Fig. 7b) results in a linear graph with a slope equal to KDE/k. From this we deduct the activation energy to be 0.45 eV. Applying EgZ 2DE, the experimental band gap size is 0.9 eV, which is larger than the calculated value. This is in agreement with the observation that the DFT-based LMTO method usually underestimates the band gap sizes [37]. The measured
3.4. Physical properties The electrical conductivity (Fig. 7a) reveals semiconducting behavior with an exponential temperature dependence, in agreement with Arrhenius’ activation model
Fig. 4. Sb–Se substructure of Pb6Sb6Se17. Thick solid: short bonds (!2. ˚ ); thin solid: intermediate bonds (2.91–2.99 A ˚ ); thin dashed: long 87 A ˚ ). bonds (3.16–3.26 A
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Fig. 5. Electronic band structures and densities of states (DOS) of Pb4Sb6S13 (top) and Pb4Sb6Se13 (bottom). The dashed horizontal lines denote the Fermi levels (EF).
electrical conductivity at room temperature is about 3.4 mS cmK1, while commercially used thermoelectrics like Bi2Te3 show conductivities in the kS cmK1 range. Due to the incongruent melting point of Pb6Sb6Se17 we were unable to obtain a phase pure sample, and physical property investigations were not performed, while we expect even smaller electrical conductivity due to its larger band gap.
4. Conclusion A new ternary semiconductor, Pb 4Sb 6Se 13 was discovered. Its crystal structure was determined by single
crystal X-ray diffraction. The cationic positions are mixed occupied by Sb and Pb atoms to different extents. The rate of the crystallization from the melt has a slight albeit significant impact on the degree of mixing. The electronic structure calculation reveals semiconducting properties, confirmed by the temperature dependent electrical conductivity measurements. The poor electrical conductivity inhibits its application as a thermoelectric material. As well, the electronic structure calculation of Pb6Sb6Se17 indicates semiconducting behavior. We are currently investigating the ternary tellurides, in which we expect smaller band gaps, hence higher electrical conductivities.
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Fig. 6. Electronic band structure and densities of states (DOS) of Pb6Sb6S17 (top) and Pb6Sb6Se17 (bottom). The dashed horizontal lines denote the Fermi levels (EF).
Fig. 7. (a) Temperature dependent electrical conductivity of Pb4Sb6Se13. (b) Logarithmic dependence of electrical conductivity versus 1/T.
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Acknowledgements Financial support from NSERC, CFI, OIT (Ontario Distinguished Researcher Award for H.K.), the Province of Ontario (Premier’s Research Excellence Award for H.K.) and the Canada Research Chair program (CRC for H.K.) is appreciated.
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Crystal and electronic structures and physical properties of two semiconductors: Pb4Sb6Se13 and Pb6Sb6Se17 Shahab Derakhshan, Abdeljalil Assoud, Nicholas J. Taylor, Holger Kleinke* Department of Chemistry, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received 26 April 2005; accepted 24 May 2005 Available online 28 July 2005
Abstract The title compounds were synthesized by directly reacting the elements in stoichiometric ratios at elevated temperatures. Their crystal structures were determined by single crystal X-ray diffraction. Pb4Sb6Se13 crystallizes in the monoclinic space group I2/m with lattice ˚ , bZ4.0910(2) A ˚ , cZ25.212(1) A ˚ , bZ93.943(1)8, VZ2530.3(2) A ˚ 3 (ZZ4), while Pb6Sb6Se17 crystallizes in dimensions of aZ24.591(1) A ˚ , bZ24.061(7) A ˚ , cZ4.1382(9) A ˚ , VZ1580.4(7) A ˚3 the orthorhombic space group P21212 with lattice dimensions of aZ15.872(4) A (ZZ2). Electronic structure calculations predicted semiconducting behavior. Temperature dependent electrical conductivity measurements verified this prediction for Pb4Sb6Se13. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Ternary alloy systems; B. Electronic structure; B. Electrical resistance; B. Thermoelectric properties
1. Introduction Thermoelectrics are materials that are able to convert thermal energy into electricity and vice versa. They are used for power generation utilizing the Seebeck effect and for heat pumping and refrigeration (Peltier effect). The applicability of these compounds is mostly limited because of their relatively low efficiency. Their efficiency depends on the dimensionless fig.-of-merit, ZT, defined as ZTZ TS2s/k. Therein, T is the absolute temperature, S is the thermopower or the Seebeck coefficient, s is the electrical conductivity, and k is the thermal conductivity [1]. Therefore, high thermopower and high electrical conductivity combined with low thermal conductivity are ideal. Since both electrons and phonons contribute to the thermal conductivity, lowering the electrical contribution, ke, will reduce and thus negatively affect s as well. The challenge for materials chemists in this field remains in the optimization of these transport properties by synthesizing materials with high S, high s and low kph. Hence, promising
* Corresponding author. E-mail address: [email protected] (H. Kleinke).
0966-9795/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2005.05.008
materials are narrow band gap semiconductors composed of heavy elements [2–4]. Binary lead, bismuth and antimony chalcogenides have been commercially used for thermoelectric applications for a long time. Moreover, the most studied materials in thermoelectric research include ternary and higher antimony chalcogenides [5–14]. In our research group we have thus far focused on potential thermoelectric materials in the metal tetrelide, pnictide and chalcogenide systems using both experimental and theoretical approaches [15–19]. Currently, we are investigating ternary and higher lead antimony chalcogenides. We recently reported on the thermoelectric properties of FePb4Sb6S14 [20]. Pb4Sb6S13 [21] is a related ternary compound (whose properties were not reported). One can assign the oxidation states of lead, antimony and sulfur to be C2, C3 and K2, respectively. A band gap may be expected to appear between fully occupied S p-states and empty Pb and Sb p-states. Replacing sulfur atoms by selenium atoms would likely shrink the band gap, in favor of higher s by elevating the occupied p-states. At the same time, the ZT value benefits from lowered phonon contribution to the thermal conductivity caused by the heavier chalcogen atom. Here we will introduce the synthesis, crystal structure, electronic band structure, as well as temperature dependent electrical conductivity of the new ternary selenide Pb4Sb6Se13. Furthermore, after we
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started our investigation in the Pb/Sb/Se system, the structures of Pb6Sb6S17 [22] and Pb6Sb6Se17 [23] were published almost simultaneously by different authors. According to these reports, they form different structures, the selenide exhibiting Se disorders within a Se2K 3 group. We therefore included Pb6Sb6Se17 in our studies, also because its properties were not reported.
2. Experimental section 2.1. Synthesis All elements were used as purchased from ALFA AESAR with purities between 99.5 and 99.9%. Single crystals of Pb4Sb6Se13 and Pb6Sb6Se17 were obtained by heating stoichiometric mixtures of the elements (overall sample mass per reaction: about 0.5 g) in evacuated quartz tubes to 600 and 800 8C, respectively. At these temperatures the products were molten, and subsequent slow cooling (2 8C/h) yielded needle like crystals. Another sample of Pb4Sb6Se13 was prepared at 600 8C with subsequent faster cooling down to room temperature by switching off the furnace. Phase pure Pb4Sb6Se13 for physical property measurements was synthesized by placing a stoichiometric mixture of Pb, Sb and Se into an alumina crucible, which was then sealed in an evacuated quartz tube and annealed at 500 8C. According to our experiments, Pb6Sb6Se17 melts incongruently, which makes it difficult to prepare pure samples. 2.2. Chemical analysis All samples were investigated by X-ray powder diffraction to check for purity by using an INEL powder diffractometer with a position sensitive detector. EDS analyses (LEO 1530, with integrated EDAX Pegasus 1200) on six selected crystals of Pb4Sb6Se13 revealed the presence of lead, antimony, and sulfur in the ratio of 17:23: 60 (in atomic percent), which compares well with the anticipated ratio of 17:26:57. 2.3. Single crystal structure studies Two black needle-shaped crystals from the Pb4Sb6Se13 samples heated to 600 8C were mounted for a room temperature data collection on a Smart Apex CCD (BRUKER), which utilizes graphite-monochromatized Mo Ka radiation. One crystal was obtained from slow cooling and the other one from faster cooling. In both cases, the crystal-to-detector distance was 4.514 cm. Data were collected by scans of 0.38 in u, for two blocks of 606 frames at fZ0 and 608. The exposure time was 60 s per frame for both crystals. A black needle-shaped crystal from the Pb6Sb6Se17 sample heated to 800 8C was mounted for a room
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temperature data collection on the same diffractometer. Data were collected by scans of 0.38 in u, for an overall of 606 frames at fZ08. The exposure time was 90 s per frame. All three data sets were corrected for Lorentz and polarization effects. Absorption corrections were based on fitting a function to the empirical transmission surface as sampled by multiple equivalent measurements using SADABS [24]. (The numerical absorption correction based on face indexing did not result in any improvement, most likely because the faces could not be determined reliably.) Diffraction peaks obtained from all frames of the reciprocal space images were used to determine the unit cell parameters by a least square analysis. For the structure refinements we employed the SHELXTL [25] program package. For Pb4Sb6Se13 the assumption of monoclinic symmetry, based on the lattice parameters, as well as the systematic absences pointed towards Pb4Sb6Se13 being isostructural to Pb4Sb6S13. The atomic positions were chosen based on the sulfide’s positions. We labeled M1–M4 as in the sulfide, and named the original M10–M15 sites M5–M10 in the selenide. Se1–Se6 correspond to S1–S6, and Se7–Se13 to S10–S16. These sites were arranged in the standard asymmetric unit. We allowed for mixed Pb/Sb occupancies on all M sites, but kept each site overall fully occupied. The decrease of the R value from 0.112 in the completely ordered model down to 0.050 in the model with mixed occupancies proves the validity of our proposed model. In some cases, the Pb/Sb mixing is at the borderline of being significant. For example, the M8 position of the slowly cooled crystal contains only 1.3(6)% Sb, which is about twice the value of its standard deviation (2s). However, the Sb content in this site in the fast cooled crystal reaches a significant value of 7.5(9)%. We list all refined occupancies for better comparison, regardless of their significance. Compared to our model for the selenide, most of the refined positions of the sulfide exhibit higher Pb contents, with the extreme being M15 with 88% Pb, corresponding to our M10 with 54% Pb. Our final refined formula for the slowly cooled crystal is Pb3.98(6)Sb6.02Se13, which is equal to 4: 6: 13 within the standard deviations. The same procedure led to Pb4.02(9)Sb5.98Se13 for the faster cooled crystal, i.e. the same formula within experimental error. On the other hand, the sulfide’s refined formula, Pb5.05Sb4.95S13, is too Pb-rich compared to the expected charge balanced formula. In case of Pb6Sb6Se17, the assumed orthorhombic symmetry was supported by the satisfying internal R value of 0.033 (based on F2). The systematic absences are in accord with the space group P21212 of the analogous sulfide. The refined Flack parameter [26] of 0.02(2) indicated that the absolute structure was correctly determined, and that the structure is not centrosymmetric, as reported for Pb6Sb6Se17 prepared via hydrothermal reaction. Again, we assigned the atoms based on the model published for the sulfide. The M atom positions were examined for mixed occupancy, and the decrease of the R value from 0.068 in the completely
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Table 1 Crystallographic data for Pb4Sb6Se13 and Pb6Sb6Se17 Empirical formula Formula weight (g/mol) Temperature (K) ˚) Wavelength (A Space group ˚) Cell dimensions, a (A ˚) b (A ˚) c (A b (8) ˚ 3) V (A No. of formula units per cell Calculated density (g/cm3) Absorp. coeff. (mmK1) F(000) Crystal size (mm) 2 Theta range Reflections collected Independent reflections (Rint) Obs. data (2s), parameter Absorption correction Min/max. transmission ratio Goodness-of-fit on F2 R(F), Rw(F2) (IO2s(I)) R(F), Rw(F2) (all data) Extinction coefficient ˚ 3) Max. diff. peak, hole (e/A
Pb3.98(6)Sb6.02Se13 2584.03 295(2) 0.71073 I2/m 24.591(1) 4.0910(2) 25.212(1) 93.943(1) 2530.3(2) 4 6.783 51.383 4302 60!3!1 3.24–70.02 13655 5925 (0.041) 4983, 150 Empirical 0.48 1.43 0.0504, 0.1008 0.0604, 0.1039 0.00036(2) 5.13, K4.52
ordered model down to 0.064 supported this assumption in general. The M2, M3, M5 and M6 sites exhibit significant Pb/Sb mixing, while the Pb content of the M4 site of 3.9(6)% may be considered insignificant. Since refining the
Pb6.12(6)Sb5.88Se17 3326.64 298(2) 0.71073 P21212 15.872(4) 24.061(7) 4.1382(9) 1580.4(7) 2 6.991 56.998 2760 70!3!2 3.08–69.98 8613 5221 (0.033) 4220, 138 Empirical 0.74 1.25 0.0643, 0.0931 0.0780, 0.0968 0.00027(2) 3.94, K4.23
occupancy factor of M1 yielded 100.7(7)% Pb, we fixed it as a Pb site, i.e. Pb1. The so-refined formula, Pb6.12(6)Sb5.88Se17, is within 2s of the charge balanced 6:6:17 ratio. Crystallographic details of these measurements are
Table 2 Atomic coordinatesa and equivalent displacement parameters of Pb4Sb6Se13 obtained via slow cooling, and occupancy factors of both cases as well as of Pb4Sb6S13 Atom
x
z
˚ 2) Ueq (A
Slow cooling occ. Pb
Fast cooling occ. Pb
Pb4Sb6S13 occ. Pb
Atom
M(1) M(2) M(3) M(4) M(5) M(6) M(7) M(8) M(9) M(10) Se(1) Se(2) Se(3) Se(4) Se(5) Se(6) Se(7) Se(8) Se(9) Se(10) Se(11) Se(12) Se(13)
0.1925(1) 0.0065(1) 0.4371(1) 0.3823(1) 0.0107(1) 0.0798(1) 0.1581(1) 0.1791(1) 0.2730(1) 0.3618(1) 0.1938(1) 0.0265(1) 0.4188(1) 0.4045(1) 0.2339(1) 0.1383(1) 0.0897(1) 0.1724(1) 0.2501(1) 0.3020(1) 0.3799(1) 0.4577(1) 0.4627(1)
0.2997(1) 0.3137(1) 0.9448(1) 0.0775(1) 0.8524(1) 0.9889(1) 0.8278(1) 0.6603(1) 0.5223(1) 0.3692(1) 0.4109(1) 0.4278(1) 0.8430(1) 0.1793(1) 0.1853(1) 0.5466(1) 0.1927(1) 0.0452(1) 0.8911(1) 0.7298(1) 0.5784(1) 0.4357(1) 0.7124(1)
0.032(1) 0.027(1) 0.029(1) 0.033(1) 0.030(1) 0.035(1) 0.028(1) 0.029(1) 0.024(1) 0.032(1) 0.018(1) 0.018(1) 0.017(1) 0.017(1) 0.021(1) 0.017(1) 0.021(1) 0.017(1) 0.016(1) 0.022(1) 0.019(1) 0.020(1) 0.028(1)
0.263(5) 0.787(6) 0.015(6) 0.036(6) 0.176(6) 0.076(6) 0.194(6) 0.987(6) 0.904(6) 0.542(6)
0.300(8) 0.721(9) 0.043(8) 0.067(9) 0.152(8) 0.172(9) 0.194(8) 0.925(9) 0.867(9) 0.582(9)
0.41 0.85 0.08 0.16 0.16 0.24 0.27 1.00 1.00 0.88
M(1) M(2) M(3) M(4) M(10) M(11) M(12) M(13) M(14) M(15) S(1) S(2) S(3) S(4) S(5) S(6) S(10) S(11) S(12) S(13) S(14) S(15) S(16)
a
All atoms on 4i, hence yZ0.
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Table 3 Atomic coordinates and equivalent displacement parameters of Pb6Sb6Se17 Atom
Wyck.
x
y
z
˚ 2) Ueq (A
Occ. Pb
Pb(1) M(2) M(3) M(4) M(5) M(6) Se(1) Se(2) Se(3) Se(4) Se(5) Se(6) Se(7) Se(8) Se(9)
4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 4c 2a
0.42210(4) 0.31945(4) 0.29823(4) 0.17250(7) 0.43883(7) 0.56345(6) 0.40671(8) 0.22781(9) 0.28859(8) 0.31492(8) 0.55067(8) 0.12016(8) 0.48651(9) 0.60512(9) 1/2
0.37208(2) 0.20317(2) 0.53522(2) 0.39546(5) 0.07004(5) 0.24314(4) 0.27801(5) 0.32950(5) 0.11378(5) 0.44606(5) 0.42535(5) 0.49148(6) 0.15130(5) 0.31407(6) 1/2
0.5050(2) 0.4950(3) 0.5111(3) 0.0339(4) 0.9947(6) 0.0054(5) 0.9993(6) 0.5216(6) 0.9868(6) 0.0200(5) 0.0177(6) 0.5141(6) 0.4812(6) 0.4738(7) 0.6868(5)
0.0296(2) 0.0253(2) 0.0262(2) 0.0340(4) 0.0507(5) 0.0364(4) 0.0162(3) 0.0213(3) 0.0172(3) 0.0153(3) 0.0168(3) 0.0224(3) 0.0210(3) 0.0233(4) 0.0149(5)
1 0.926(6) 0.901(6) 0.039(6) 0.098(7) 0.098(6)
summarized in Table 1. The atomic coordinates, occupancy factors as well as thermal displacement values may be found in Tables 2 and 3. 2.4. Electronic structure calculations We carried out self-consistent tight-binding LMTO calculations (LMTO: linear muffin tin orbitals) [27,28]. In this method, the density-functional theory is applied with the local density approximation (LDA) [29]. The required crystallographic data set for Pb4Sb6S13 was obtained from Skowron et al. The 10 M sites of both Pb4Sb6S13 and Pb4Sb6Se13 were chosen as Sb1, Pb2, Sb3, Sb4, Sb5, Sb6, Sb7, Pb8, Pb9, and Pb10. In this model, Sb rich sites were treated as pure Sb and the rest as pure Pb sites. The same
Fig. 1. Crystal structure of Pb4Sb6Se13 in projections along the b axis. Horizontal: a axis. Small, white circles: Se; gray: Pb/Sb. The larger the Pb content, the larger and darker the circle.
argument was applied for Pb6Sb6Se17, where we assigned Pb1, Pb2, Pb3, Sb4, Sb5, and Sb6 to the respective M sites. Thereby our models have the charge balanced elemental ratios (4:6:13 and 6:6:17). The integration in k space was Table 4 ˚ ) of Pb4Sb6Se13 (slow cooling) Selected bond distances (A M(1)–Se(5) M(1)–Se(1) M(1)–Se(5) M(1)–Se(4) M(2)–Se(2) M(2)–Se(4) M(2)–Se(3) M(2)–Se(13) M(3)–Se(3) M(3)–Se(6) M(3)–Se(2) M(4)–Se(4) M(4)–Se(1) M(4)–Se(2) M(5)–Se(7) M(5)–Se(13) M(5)–Se(12) M(6)–Se(8) M(6)–Se(11) M(6)–Se(12) M(7)–Se(9) M(7)–Se(10) M(7)–Se(11) M(8)–Se(6) M(8)–Se(9) M(8)–Se(3) M(8)–Se(10) M(8)–Se(10) M(9)–Se(11) M(9)–Se(8) M(9)–Se(9) M(9)–Se(1) M(9)–Se(6) M(10)–Se(12) M(10)–Se(7) M(10)–Se(8) M(10)–Se(5)
2x
2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x
2x 2x 2x
2.739(1) 2.804(1) 3.124(1) 3.214(1) 2.884(1) 2.993(1) 3.097(1) 3.339(1) 2.574(1) 2.778(1) 3.055(1) 2.586(1) 2.801(1) 3.0448(1) 2.646(1) 2.725(1) 3.267(1) 2.600(1) 2.877(1) 2.982(1) 2.678(1) 2.729(1) 3.309(1) 2.972(1) 3.034(1) 3.157(1) 3.387(1) 3.450(1) 2.898(1) 3.032(1) 3.073(1) 3.306(2) 3.411(8) 2.799(1) 2.883(1) 3.129(1) 3.341(1)
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˚ ); thin solid: intermediate bonds (2.88–2.98 A ˚ ); thin dashed: long bonds (3.26–3.31 A ˚ ). Fig. 2. Sb–Se substructure of Pb4Sb6Se13. Thick solid: short bonds (!2.80 A
performed by an improved tetrahedron method [30] on grids of 1098, 1098, 228 and 228 independent k points of the first Brillouin zone for Pb4Sb6S13, Pb4Sb6Se13, Pb6Sb6S17 and Pb6Sb6Se17, respectively. The first Brillouin zone (of the primitive reciprocal cell) contains two formula units in each case. 2.5. Physical property measurements We pressed part of the ground phase-pure sample of Pb4Sb6Se13 into a bar-shaped pellet of the dimensions 6!1!1 mm for the temperature dependent electrical conductivity measurements using a four-point-method. A home-made device was used to measure the voltage drops DV over a distance of 2 mm under dynamic vacuum between 275 and 325 K, wherein cooling was achieved by helium compression. Silver paint (TED PELLA) was used to create the electrical contacts.
3. Results and discussion 3.1. Crystal structure of Pb4Sb6Se13 Pb4Sb6Se13 is isostructural with Pb4Sb6S13. The structure (Fig. 1) is composed of ribbons of edge-sharing distorted square-pyramidal MSe5 groups. The M5, M6 and M7 atoms are located in a ribbon with a width of three edge-sharing square-pyramids. The M3 and M4 atoms form another ribbon with a width of two edge-sharing square-pyramids. Each of these five M sites possesses over 80% Sb content. The other M–Se chains interconnect these one-dimensionally extended motifs. Pb atoms dominantly occupy the latter sites. The coordination numbers of the M sites as well as the M–Se distances are directly related to the Pb content in the
positions (Table 4). The sites with Pb content over 50% have coordination numbers of seven and eight. The environment of the M3 and M4 atoms, almost exclusively occupied by Sb atoms, is comparable to the Sb atom within the ribbons of Sb2Se3 [31]. In this binary compound, one of ˚ to the Se atom the Sb atoms forms one short bond of 2.59 A ˚) at the apex of the pyramid, and two intermediate (2.80 A ˚ ) to the Se atoms within the and two long bonds (3.01 A plane of pyramid, i.e. overall a coordination number of 5. Neglecting the two longer bonds, one can describe the structure based on corner-sharing SbSe3 trigonal pyramids, which are interconnected along the b direction. The corresponding trigonal pyramids of Pb4Sb6Se13 are depicted in Fig. 2 by the thick solid lines. The M8 site, occupied by 99% Pb in the slowly cooled crystal, is in an eight-coordinated environment with M–Se ˚ . Such a coordination distances between 2.97 and 3.45 A sphere for Pb is present, for instance, in the crystal structure of PbU2Se5 where eight Pb–Se bond distances are in the ˚ [32]. The M10 atom exhibits range between 2.95 and 3.50 A considerable mixing with 54% Pb and 46% Sb. There are seven M10–Se bonds per M10 atom with distances between ˚ , which are comparable to the M3 atom in 2.80 and 3.34 A PbSb2Se4 [33] (58% Pb content) with seven M3–Se bonds ˚ ). (2.77–3.39 A Since the Pb/Sb ratio is 4:6, statistics dictates that 60% of each of the 10 cation sites will be occupied by Sb. This would be expected, were the entropy the dominating factor. The sites with less than 60% Sb atoms occupancy in the slowly cooled crystal have higher Sb contents in the faster cooled crystal. On the other hand, the sites with more than 60% Sb occupancy in the slowly cooled crystal exhibit lower Sb contribution in the faster cooled crystal. Hence faster cooling yielded a more statistical distribution, i.e. partial ordering occurs during the cooling process. Using the LMTO method we calculated the muffin tin radii for these 10 cation positions. To avoid
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Table 5 ˚ ) vs. Sb occupancies of the M sites of Pb4Sb6Se13 The muffin tin radii (A Site
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
˚) Radius (A Sb slow cooling (%) Sb fast cooling (%)
1.656 0.737(5) 0.700(8)
1.746 0.213(6) 0.279(9)
1.553 0.985(6) 0.937(8)
1.561 0.964(6) 0.933(9)
1.598 0.824(6) 0.848(8)
1.568 0.924(6) 0.828(9)
1.617 0.806(6) 0.806(6)
1.841 0.013(6) 0.075(9)
1.761 0.096(6) 0.133(9)
1.692 0.458(6) 0.418(9)
the effect of the atomic size on the site radius, we assumed all the sites are fully occupied by the Sb atoms. The results are summarized in Table 5. Increasing size corresponds to decreasing Sb contents, as expected based on the smaller size of the Sb atom, compared to Pb.
distances between the two kinds of sites, the Pb/Sb distribution is much more ordered in Pb6Sb6Se17 than in Pb4Sb6Se13, as each site is clearly dominated (by 90% or more) either by Pb or Sb atoms. 3.3. Electronic structures
3.2. Crystal structure of Pb6Sb6Se17 The structure of Pb6Sb6Se17 was originally reported [23] in the space group Pbam. In this model there is a ˚ between two Se9 problematic short contact of 1.54 A atoms. Therefore, the occupancy of Se9 was fixed at 0.5 assuming local ordering that avoids this physically impossible Se–Se distance. Our crystal structure investigation revealed that this compound is isostructural with Pb6Sb6S17, where a model in the space group P21212 with complete S ordering was applied. Hence, our Pb6Sb6Se17 crystallizes in an ordered variant of the hydrothermally prepared Pb6Sb6Se17. The Wyckoff notation of all the atoms is 4c except for the Se9 (2a). Moreover, in contrast to the previously reported model, we noticed that except for Pb1 all the cation sites are mixed occupied by Pb and Sb, with up to 10% Sb on Pb and vice versa (Table 3). The mixing in the M4 site is about 6s and may be insignificant. The structure of Pb6Sb6Se17 (Fig. 3) is composed of ribbons of the width of three edge-sharing distorted squarepyramidal MSe5 groups, which run along the shortest crystallographic axis, c, and Pb rich sites connect these ribbons. An additional feature in the structure of this compound is the presence of the Se2K 3 unit formed by one Se9 atom and two Se5 atoms. Hence the formula may be written as ðPb2CÞ6 ðSb3CÞ6 ðSe2KÞ14 ðSe3 Þ2K. This selenide is structurally related to Sr6Sb6S17 [34]. The unit cell in the latter compound is doubled along the short axis compared to Pb6Sb6Se17, and the space group is P212121. The difference between these two stems from the Sb–Q substructures: the Sb atoms of Pb6Sb6Se17 are aligned in the same line along the shortest axis c (Fig. 4) because of translational symmetry, while the Sb atoms of the Sr compounds form a zigzag chain, thereby causing the cell doubling. Five Se atoms form the coordination spheres of the ˚, positions with high Sb contents (Table 6: 2.57–3.26 A ˚ as a cutoff). Seven and eight Se atoms surround using 3.5 A ˚ ). This correlates nicely to the Pb rich positions (2.93–3.43 A the well-known tendency of the Pb atoms to prefer sites with larger distances and higher coordination numbers. Most likely because of larger differences in the interatomic
The densities of states (DOS) and the band structures of Pb4Sb6S13 and Pb4Sb6Se13, both shown in Fig. 5, are indicative of semiconducting behavior. The Fermi level is located in a gap between fully occupied states of chalcogen Table 6 ˚ ) of Pb6Sb6Se17 Selected bond distances (A Pb(1)–Se(1) Pb(1)–Se(1) Pb(1)–Se(5) Pb(1)–Se(4) Pb(1)–Se(5) Pb(1)–Se(8) Pb(1)–Se(2) Pb(1)–Se(4) Pb(1)–Se(9) M(2)–Se(7) M(2)–Se(3) M(2)–Se(3) M(2)–Se(1) M(2)–Se(1) M(2)–Se(9) M(2)–Se(8) M(3)–Se(4) M(3)–Se(6) M(3)–Se(4) M(3)–Se(3) M(3)–Se(3) M(3)–Se(5) M(3)–Se(5) M(3)–Se(9) M(4)–Se(4) M(4)–Se(2) M(4)–Se(2) M(4)–Se(6) M(4)–Se(6) M(5)–Se(3) M(5)–Se(7) M(5)–Se(6) M(5)–Se(6) M(5)–Se(7) M(6)–Se(1) M(6)–Se(8) M(6)–Se(8) M(6)–Se(7) M(6)–Se(7) Se(5)–Se(9)
2x
3.060(2) 3.092(2) 3.142(2) 3.177(2) 3.211(2) 3.225(2) 3.250(2) 3.256(2) 3.401(1) 2.931(2) 3.001(2) 3.048(2) 3.061(2) 3.085(2) 3.371(2) 3.430(2) 2.967(2) 3.016(2) 3.016(2) 3.117(2) 3.129(2) 3.289(2) 3.325(2) 3.392(2) 2.568(2) 2.714(3) 2.790(3) 3.159(2) 3.264(2) 2.607(2) 2.906(3) 2.929(3) 2.980(3) 2.985(3) 2.626(2) 2.666(3) 2.862(3) 3.202(3) 3.229(2) 2.398(2)
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Fig. 3. Crystal structure of Pb6Sb6Se17 in a projection along the c axis. Horizontal: b axis. Small, white circles: Se; gray: Pb/Sb. The larger the Pb content, the larger and darker the circle.
p-states and empty Sb and Pb p-states. The band gaps in both compounds are indirect. The computed gap size in Pb4Sb6S13 is about 0.8 eV, and in the selenide 0.55 eV. The smallest direct gap size are about 0.9 eV for the sulfide and 0.65 eV for the selenide. The anticipated decrease of the gap size is caused by the valence p-states of the selenide being located at higher energies. The densities of states (DOS) and the electronic band structure of Pb6Sb6S17 and Pb6Sb6Se17 are shown in Fig. 6. Semiconducting properties are to be expected here as well. Indirect band gaps of 1.05 and 0.7 eV and direct band gaps of 1.2 and 0.8 eV are observed for the sulfide and selenide, respectively. The partial DOS calculations prove that the Q atoms dominate the occupied states (in addition to the Sb s-states). It is assumed that the Pb4Sb6Q13 family exhibits smaller gaps than Pb6Sb6Q17 because of the smaller Pb/Sb ratio (due to the lower electronegativity of Pb). We analyzed the Crystal Orbital Hamilton Populations (COHP curves) [35] of selected interactions involving Q5 and Q9 of Pb6Sb6S17 and Pb6Sb6Se17 to investigate the effect of the Q–Q bonds on the band gap size. Since the antibonding states of the Q–Q interactions overlap with the empty Sb p-states, hence occur at the same energies, their impact on the gap size cannot be huge.
ðsZ s0 expðKDE=kTÞÞ, with kZBoltzmann constant and DEZactivation energy [36]. Plotting the electrical conductivity logarithmically as a function of 1/T (Fig. 7b) results in a linear graph with a slope equal to KDE/k. From this we deduct the activation energy to be 0.45 eV. Applying EgZ 2DE, the experimental band gap size is 0.9 eV, which is larger than the calculated value. This is in agreement with the observation that the DFT-based LMTO method usually underestimates the band gap sizes [37]. The measured
3.4. Physical properties The electrical conductivity (Fig. 7a) reveals semiconducting behavior with an exponential temperature dependence, in agreement with Arrhenius’ activation model
Fig. 4. Sb–Se substructure of Pb6Sb6Se17. Thick solid: short bonds (!2. ˚ ); thin solid: intermediate bonds (2.91–2.99 A ˚ ); thin dashed: long 87 A ˚ ). bonds (3.16–3.26 A
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Fig. 5. Electronic band structures and densities of states (DOS) of Pb4Sb6S13 (top) and Pb4Sb6Se13 (bottom). The dashed horizontal lines denote the Fermi levels (EF).
electrical conductivity at room temperature is about 3.4 mS cmK1, while commercially used thermoelectrics like Bi2Te3 show conductivities in the kS cmK1 range. Due to the incongruent melting point of Pb6Sb6Se17 we were unable to obtain a phase pure sample, and physical property investigations were not performed, while we expect even smaller electrical conductivity due to its larger band gap.
4. Conclusion A new ternary semiconductor, Pb 4Sb 6Se 13 was discovered. Its crystal structure was determined by single
crystal X-ray diffraction. The cationic positions are mixed occupied by Sb and Pb atoms to different extents. The rate of the crystallization from the melt has a slight albeit significant impact on the degree of mixing. The electronic structure calculation reveals semiconducting properties, confirmed by the temperature dependent electrical conductivity measurements. The poor electrical conductivity inhibits its application as a thermoelectric material. As well, the electronic structure calculation of Pb6Sb6Se17 indicates semiconducting behavior. We are currently investigating the ternary tellurides, in which we expect smaller band gaps, hence higher electrical conductivities.
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Fig. 6. Electronic band structure and densities of states (DOS) of Pb6Sb6S17 (top) and Pb6Sb6Se17 (bottom). The dashed horizontal lines denote the Fermi levels (EF).
Fig. 7. (a) Temperature dependent electrical conductivity of Pb4Sb6Se13. (b) Logarithmic dependence of electrical conductivity versus 1/T.
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Acknowledgements Financial support from NSERC, CFI, OIT (Ontario Distinguished Researcher Award for H.K.), the Province of Ontario (Premier’s Research Excellence Award for H.K.) and the Canada Research Chair program (CRC for H.K.) is appreciated.
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