Reconciling Crystallographic and Physical Property Measurements on

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Reconciling crystallographic and physical property measurements on thermoelectric lead sulphide Christian Moeslund Zeuthen, Peter Skjøtt Thorup, Nikolaj Roth, and Bo Brummerstedt Iversen J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.9b00043 • Publication Date (Web): 01 May 2019 Downloaded from http://pubs.acs.org on May 1, 2019

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Journal of the American Chemical Society

Reconciling crystallographic and physical property measurements on thermoelectric lead sulphide

Christian Moeslund Zeuthen, Peter Skjøtt Thorup, Nikolaj Roth and Bo Brummerstedt Iversen*

Center for Materials Crystallography, Department of Chemistry and iNANO, Aarhus University, Langelandsgade 140, Aarhus, Denmark.

ABSTRACT For many decades the lead chalcogenides PbTe, PbSe and PbS (and their solid solutions) have been preferred high performance thermoelectric materials due to their exceptional electronic and thermal properties as well as great stability during operation. However, there is a lack of understanding about the fundamental relation between the reported highly defect crystal structure containing cation disorder and vacancies, and the observed transport properties, which follow expectations for an ideal rock salt crystal structure. Here we have studied a series of undoped lead sulfide samples (Pb1-xS) with presumed small chemical variations. Crystallographic refinements of high resolution synchrotron powder X-ray diffraction data give unphysically low lead occupancies (0.75-0.98), in contradiction with the measured charge carrier concentration, resistivity, mobility and Seebeck coefficient that show no signs of lead vacancies. A new Rietveld refinement model including preferred orientation parameters and anisotropic strain gives almost full lead occupancy and improved agreement factors. However, Transmission Electron Microscopy analysis reveals that there is no preferred orientation in this

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system. Instead it is the diffuse scattering due to directional correlated disorder in the structure that necessitates the additional parameters when modelling Bragg intensities. The present approach is a general method for absorbing effects of direction dependent correlations in advanced materials.

INTRODUCTION One of the most important groups of semiconductor materials is the lead chalcogenides (PbTe, PbSe and PbS), which like silicon have entire book volumes dedicated to their structures and properties.1, 2 The lead chalcogenides are used in many fields such as optoelectronic devices,3,4 photoemitters,5 and very importantly as intermediate temperature thermoelectric materials.6 The wide applicability is both due to the extraordinary properties, and the widespread abundance with PbS being found naturally as the mineral Galena. In fact PbS was used in the first thermoelectric devices.1 In addition, the synthesis of lead chalcogenides is quite straightforward even on the nanoscale. Due to the extraordinary stability, malleability, availability and effeciency, PbTe is used in modern thermoelectric devices, and there are hundreds of recent studies focusing on further enhancing the thermoelectric figure of merit in lead chalcogenides through careful doping and control of the hierarchical structure.7–15 All the lead chalcogenides crystallize in the high symmetry rock salt structure (𝐹𝑚3𝑚), which is in itself interesting since many high performance thermoelectric materials tend to crystallize in structures with large unit cells such as clathrates, Zintl compounds and skutterudites.16–20 These compounds all display low thermal conductivities attributed to their complex unit cells, and in some cases also the presence of loosely bonded “rattler” atoms.21, 22 The lead chalcogenides surprisingly also have low thermal conductivities reaching down to 2.26. W/mK,23 even though they have highly symmetric small unit cells. The origin of this low thermal conductivity is still debated with one explanation being local symmetry breakage due


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to the lead atoms being displaced away from the high symmetry rock salt lattice position. Whether the displacement is a static or dynamic is still not fully understood,24–27 but a recent 3D-ΔPDF and Molecular Dynamics (MD) study suggested correlated local dipole formation in PbTe via an anion-cation dimerization, which dynamically displaces the lead atoms.28 Other studies attribute the low thermal conductivity to large anharmonic interactions between the longitudinal acoustic and transverse optic phonon modes, which results in a high degree of phonon-phonon scattering.29 Unravelling the details of the crystal structures of lead chalcogenides are at the forefront of structural science. It is, however, also interesting to note that even after hundreds of studies of lead chalcogenides, there is not agreement about the basic physical properties, such as electrical (𝜎) and thermal (𝜅) conductivity and Seebeck coefficient (𝑆), and thus also disagreement about the figure-of-merit, 𝑧𝑇 =

𝑆2𝜎 𝜅 𝑇

with T being the absolute temperature. Most

studies concern doped lead chalcogenides, as these have optimal carrier concentration.7,9,37– 39,29–36 For undoped lead chalcogenides, there are very large differences in the reported electrical

properties. The electrical conductivity decreases significantly when samples are thermally cycled, i.e 0.2 Ω 𝑐𝑚 to 0.002 Ω 𝑐𝑚 at 300 K after four thermal cycles, while the carrier concentration changes one order of magnitude from 4 x 1017 cm-3 to 7 x 1018 cm-3.40 Another study revealed that the majority carrier type is n-type in Pb-rich environments, and p-type in S, Se or Te rich environments as well as the ability to switch from one carrier type to the other by heating the system in an atmosphere of either Pb or S.41, 42 All these observations suggest that small chemical deviations from the nominal stoichiometry have significant influence on the physical properties. To understand the basic chemistry it is therefore necessary to first examine the pure materials without dopants, and this has recently led to a re-emergence of the wellestablished phase boundary mapping analysis.43 All lead chalcogenides are reported as being close to line phases, with only small deviations from 1:1 atomic ratio of the elements. In fact,



the lead chalcogenides are reported as stoichiometric in almost all studies. In contrast, Christensen et. al. reported, based on Rietveld refinements of synchrotron powder X-ray diffraction (PXRD) data, the presence of significant amounts of lead vacancies.25 The defects were most pronounced in PbS, where the lead occupancy vary from 96 - 98 %. That study also suggested significant structural differences between two PbS samples despite that the synthesis method was almost identical. Similar observations have been reported in the Quantum Dot (QD) PbS materials, where a surprising amount of lead vacancies (up to 10 %) was reported.44 Deficiency of lead in the structure must lead to a large change in the carrier concentration, and thus a large change in the electrical properties of the materials. This means that small variations in the synthesis of pure lead chalcogenide potentially could explain the discrepancies in physical properties reported in literature. While single crystals could give more insight into the true structure of the lead chalcogenides, this paper attempts to identify and reconcile differences between physical property measurements and crystallographic analysis with PXRD as this is the method most often used in literature. Furthermore, single crystals may not be representative for the powder samples studied by physical property measurements. Here we report a combined structural, physical and electrical properties study of PbS synthesized with small stoichiometric variations in the synthesis method to imitate the variation in preparation methods in the literature. PbS was chosen among the lead chalcogenides since it has the largest reported deviation from full site occupancy as well as most significant lead offcentering.17 To ensure the optimal comparability between physical and crystallographic properties, the studied samples should be identical. Therefore powders from the same synthesis batch are used both to press pellets and for PXRD data collection.

EXPERIMENTAL SECTION Synthesis


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To produce the samples, the desired amounts of lead (99.99 %) and sulphur (99.9 %) were weighted in air, and then transferred to fused silica tubes coated with a carbon layer produced by pyrolysis of acetone. The tubes were subsequently evacuated using a vacuum pump (below 5 x 10-4 mbar) and flame-sealed with a torch to create an ampoule. The samples were slowly heated at a rate of 100 oC/h to 400 oC, and held for 1 hour to prevent sulphur evaporation from the melt. Then the samples were heated at a rate of 250 oC/h to 1150 oC followed by an isothermal plateau for three hours and finally freely cooled. The resulting ingots were ground roughly by hand and ball-milled for five minutes with tungsten carbide balls to ensure similar grain sizes. Each powder was compacted by spark plasma sintering (SPS) (SPS-515, SYNTEX Inc, Japan) to a 1-inch diameter pellet. In the SPS press, the powders were quickly heated at a rate of 50 oC/h to 500 oC and held for 10 minutes under a pressure of 60 MPa. The density of the resulting pellets were measured, by the Archimedes method, to be >97% of the theoretical density. The pellets were subsequently sliced to appropriate dimensions for analysis using a wire saw.

Powder X-ray diffraction PXRD data was collected at beamline BL44B2 at the SPring-8 synchrotron facility in Japan. These high-resolution data were collected on an image plate detector with wavelengths of 0.499815(10) Å in December 2016, 0.500279(3) Å in July 2017, and 0.499843(11) Å in December 2017, respectively. All data were measured at 100 K unless otherwise noted. The wavelengths were determined by Rietveld refinement of data measured on a LaB6 standard (NIST 660b) and/or a CeO2 standard. Powdered samples were floated in ethanol, dried and packed in glass capillaries for low temperature measurement, and in fused silica capillaries for high temperature measurements (Ø = 0.1 or 0.2 mm). Random distribution and dense packing of particles were ensured using an ultrasonic bath during the capillary loading process.



Crystallographic refinements The PXRD data were first fitted using the Le Bail method in JANA2006 to determine the peak shape.45 Manual background points were chosen and interpolated using a 5-term Chebyshev polynomial function. Slight sample displacements were corrected for using the systematical cos(θ) dependent shift parameter. The Thompson-Cox-Hastings pseudo-Voigt function was used to describe the peak shape including the Simpson correction for asymmetry.46 For LaB6, the values of U, W, X and Y were refined, while for the PbS data, the U and X values were constrained to values obtained for the LaB6 data, and only the W and Y parameters were refined. Strain in the powder was refined using the Stephens model with three parameters, 𝜁, 𝑆400 and 𝑆220.47 Additionally, some refinements included the March-Dollase model for preferred orientation, refining only the March parameter.48 Atomic positions were introduced based on the 𝐹𝑚3𝑚 space group with NaCl structure. For Pb, anharmonic atomic displacement parameters (ADPs) were refined using a 4th order Gram-Charlier expansion, while S was refined with isotropic ADP. The packing density in the capillaries was estimated to be 50 % of the theoretical density, which resulted in an estimated µR of 1.55 for Ø = 0.2 mm and 0.78 for Ø = 0.1 mm for the absorption correction. Single crystal X-ray diffraction measurements were carried out at 100 K on a Bruker Kappa APEXII diffractometer equipped with an Incoatec microfocus Ag tube (wavelength 0.56 Å) and a CCD detector. The data was integrated and corrected for absorption in the APEXII software suite. The structure solution and refinement were carried out with SHELXS and SHELXL, respectively, using the Olex2 software.49–51

Physical measurements


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All physical property measurements were carried out both at high-temperatures (300 K to 825 K) and low temperatures (5 K to 100 K). All measurements were carried out on different sample pieces cut from the same pellet to assure that there is no history, i.e. that all measurements were independent of previous measurements or small variations when pressing several different pellets. An illustration of the pellet geometry and the cutting process can be found in the supporting information. The homogeneity of the pressed pellets before cutting samples for physical property measurements was evaluated with a PANCO Potential-Seebeck-Microprobe (PSM). The local Seebeck coefficient, both radially and for the cross-sections of the pellet, is a good test for homogeneity, as small variations in the local Seebeck value stem from local chemical alteration. The spatial resolution of the PSM is 100 x 100 m. Thermal diffusivity, 𝐷, was measured using the laser flash method (LFA) with a Netzsch LFA 457 equipped with an InSb detector. The pressed pellets were polished and cut to Ø = 12.7 mm to fit the LFA sample holder. All samples were coated with a thin layer of graphite (Graphit33, Kontakt Chemie) to minimize errors from reflection of the laser pulse. The effect of the graphite layer on the thermal diffusivity is assumed negligible. To provide an inert atmosphere, all measurements were performed a 5 mL/min Ar flow. The pulse signals were analyzed with the Cape-Lehmann model using a pulse-width correction, which takes heat-loss through radiation and finite pulse-time effects into account. Thermal conductivity, 𝜅, was calculated from 𝜅 = 𝐷𝑑𝐶𝑝, where 𝑑 is the thickness of the pellet and 𝐶𝑝 is heat capacity. 𝐶𝑝 was measured against a reference sample (Pyroceram 9606) of known density and heat capacity.52 The observed heat capacities were compared with the Dulong-Petit value and found to correspond well with these. Resistivity and charge carrier concentration measurements were performed on a homebuild Hall setup described by Borup et al.,53 which uses the Van-der-Pauw method. For Hall



measurements, a 1.25 T magnetic field was applied to samples in vacuum (< 1 x 10-4 bar). The samples were mounted so that the resistivity is measured perpendicular to the SPS pressing direction. The Seebeck coefficient was measured on a home-build setup similar to the one described by Iwanaga.54 The data were measured with the temperature gradient parallel to the SPS pressing direction. From 300 K to 425 K, the data were measured in ambient air, while the data from 425 K to 825 K were measured in vacuum (< 1 x 10-4 bar). Transmission Electron Microscope (TEM) images were collected on a FEI Talos F200X operated in TEM bright field mode at 200 kV. Samples were prepared by suspending a fine powder in ethanol, and applying the suspension to TEM grids (TED Pella inc. carbon foil on 200 mesh cobber).

RESULTS AND DISCUSSION Crystallography Synchrotron PXRD measurements were performed at 100 K on the Pb1-xS powder samples (x = 0.00, 0.01, 0.02, 0.04) as well as on powders obtained from crushing single crystals. The samples are listed in Table 1, and they were all treated the same way prior to capillary packing. Three identical samples were produced (PbS-01, PbS-03 and PbS-04) to identify any small changes resulting from similar synthesis methods. These three samples were prepared in separate fused silica tubes, and all three were placed simultaneously in the same oven. Data were measured during three different beamtimes to diversify the crystallographic measurements, and Figure 1 shows the measured diffractograms. PbS is the main phase, but in some samples, very small amounts of Pb and PbO (Litharge) are found as additional phases although the contents are too small to refine (estimated < 1 %). Such minute amounts of impurity phases will typically not be resolved in laboratory based PXRD data.


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Figure 1. PXRD data at 100 K with sample nomenclature as in Table 1. The sample “Z-PbS03 + CeO2” is a mixture of PbS and CeO2 (NIST SRM 674b) and it was used for accurate determination of the unit cell parameter. The expected reflection positions for PbS and CeO2 are also shown. The data are shown in log scale to enhance potential impurity phases.

Table 1. Formal stoichiometry, lead occupancy and agreement factors obtained from Rietveld refinement of PXRD data. PbS-03/04-SC consist of crushed single crystals obtained during the synthesis of PbS-03/04. Data were recorded at 100 K. Sample name

Formal Stoichiometry

PbS-01

wRp (%)

Pb1S

Unit Cell Pb Occupancy Parameter (Å) 5.9116(1) 0.965(3)

PbS-03

Pb1S

5.9166(2)

0.969(3)

2.67

PbS-03-SC

Pb1S

5.9151(2)

0.985(4)

3.17

PbS-04-SC

Pb1S

5.9154(2)

0.972(5)

2.57

PbS-04

Pb1S

5.9139(2)

0.914(3)

2.62

PbS-06

Pb0.99S

5.9168(2)

0.944(3)

2.93

PbS-07

Pb0.98S

5.9163(2)

0.984(3)

2.87

PbS-08

Pb0.96S

5.9149(2)

0.974(3)

3.00


3.80


Elaborate Rietveld refinements on high quality synchrotron PXRD data on lead sulfide previously has shown significant off-stoichiometry as well as differences between samples.25 In the present study a similar range of lead vacancies between 1-9 % is observed as listed in Table 1. There is no clear relation between the formal stoichiometry and the refined amount of vacancies. Small single crystals were recovered from the quartz tubes used in the synthesis of sample PbS-03. Single crystal X-ray diffraction data were measured on three different crystals and site vacancies were refined to be 2.0(8) %, 2.4(8) % and 2.2(7) %, respectively. This is close to the amount found by the Rietveld refinement of the PbS-03 powder (3.1(3) %). In addition, single crystals from the same batch were crushed and Rietveld refinement on these powder data gave 1.5(4) % lead vacancies. Interestingly, the three identically synthesized samples (PbS-01, 03 and 04), showed rather large differences in both the unit cell parameters and occupancy, warranting further careful investigation. In literature, defect formation in lead chalcogenides has been studied both by ab initio theory55 and by experiment.56,57 It has been suggested that the primary point defect in chalcogenide-rich conditions is lead vacancy formation where the equilibrium concentration of lead vacancies in PbS at 298 K was calculated to be 7.47 x 10-31,55 which is much below the 1-9 % found here. From experimental studies of the lead-sulphur system, a very small solidsolution range of just 0.08 atomic percent was found at 900 oC,58 but this value is somewhat disputed by other sources, by up to one order of magnitude.59 While no direct experimental research of the lead vacancies in PbS has been reported, the vacancy defects in the isostructural PbSe and PbTe has been studied by positron annihilation, where it was shown that the vacancies are responsible for the increase in charge carrier concentration, caused by either VPb** or VPb*.56, 57 Assuming that all lead vacancies are charged,


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we would expect the unit cell length to decrease with increasing charge carrier concentration (increasing vacancy concentration). This relation between charge carrier concentration and crystallographic properties has previously been examined in the PbTe system, where a decreasing trend of the lattice parameters as a function of carrier concentration was reported. However, on further review, the results seems to be inconclusive as the proposed trend lines do not match well to the actual data.59

Figure 2. Unit cell parameter at 100 K as a function of lead occupancy. The dotted line is a least-squares fit to a straight line. The data may be affected by slight temperature variations in the cooling setup at the beamline, as data were measured at three different beamtimes.

Here, the unit cell lengths were determined from the synchrotron PXRD data, and they are plotted as a function of occupancy in Figure2. Changes in unit cell length as a function of refined occupancy are quite small, on the order of 10-3 Å, and no clear trends are seen. In other compounds, deficiency often results in changes in the unit cell parameters on the order of 10-1 Å.60,

61

Considering the changes in refined lead vacancies and the deviations in unit cell

parameter, the correlation looks very weak. Either the refined lead occupancy is inaccurate or the previous hypothesis that the unit cell parameter is decreasing with increasing vacancy



concentration is wrong. The small differences between the samples can also originate from slight differences in sample temperature, as the different data sets were measured during three different beam times. Overall, the unit cell parameter does not change significantly between the samples suggesting that they are indeed similar to each other with little variation in lead occupancy. The unit cell parameter is very well determined in Rietveld refinements, while the occupancy can have larger errors. This is demonstrated by a temperature series measured for sample PbS-05 (Figure S4 in the supporting material). From these data, the linear thermal expansion of PbS at 300 K is estimated to be 2.19(1) × 10 ―5 Å ―1, which is quite constant throughout the entire temperature series. This trend is the same as previously found in literature,26,62,63 whereas the absolute value is slightly higher previous studies. This can be explained by the omission of low temperature data, which has a slightly lower thermal expansion.63 PbS decomposes irreversibly in air to Lanarkite, PbO•PbSO4, beginning at 700 K as previously described by Binnie.64 Below we will further explore the link between lead occupancy and unit cell parameter through measurement of the charge carrier concentrations to identify the source of inconsistency between these quantities.

Physical properties Several heating and cooling cycles were measured for all physical properties in order to probe potential annealing or degradation effects. Figure 3 shows examples of several heating cycles for both  and S. The first heating cycle shows clear irreversible behavior in both properties. This may be due to grain boundary annealing or due to sulphur evaporation as previously seen in this system.40 Another explanation is that the samples are not in chemical equilibrium during the first heating cycle. After the first heating cycle, the properties do not change further with subsequent cycles. This indicates that there is no continuous degradation of the material and that the samples have now obtained chemical equlibrium. In the resistivity, there is a hysteresis


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behavior around the maximum value. It is clear that this is a reversible effect but the origin is unclear. In all following plots of physical properties, the samples have been cycled a minimum of two times, but only the last cooling cycle is shown.

Figure 3. Resistivity and Seebeck coefficient for PbS-06 during three heating and cooling cycles.

Table 2. Hall carrier concentration, resistivity and Hall mobility values at 50°C. The effective mass is calculated using the single band Kane model with both acoustic and optical phonon scattering and is given in units of electron mass (𝑚𝑒). Sample

Hall

Name

carrier Resistivity

Hall

Seebeck

Effective

concentration (𝒎𝛀 𝒄𝒎)

mobility

(𝝁𝑽 𝑲 ―𝟏)

Mass

(𝟏𝟎 ―𝟏𝟖 𝒄𝒎 ―𝟑

(𝒄𝒎𝟐 𝑽 ―𝟏

)

𝒔 ―𝟏)

(𝒎𝒆)

PbS-04

―2.9971(12)

7.114(4)

292.85(12)

―162(1)

0.176

PbS-06

―1.8808(5)

10.531(7)

315.14(8)

―219(2)

0.210

PbS-07

―2.3609(19)

8.147(5)

323.14(30)

―250(1)

0.324

PbS-08

―2.6443(13)

6.451(2)

365.88(22)

―221(1)

0.276



Charge carrier concentration, resistivity and mobility The charge carrier concentrations, 𝑛, of each sample was measured and can be seen in Figure 4a, while values obtained at 50 ℃ are listed in Table 2. The carrier concentration varies only on the order of 1 x 1018 cm-3 across the samples, whereas any changes coming from offstoichiometry are expected to lead to much greater variation. Assuming that all vacancies are doubly charged, the corresponding change in charge carrier concentration can be calculated as 𝑛𝑣 = 2 ∗

𝐶𝑣

#(1) 𝑉𝑈.𝐶 4

where 𝑛𝑣 is the charge carrier concentration from vacancies, 𝐶𝑣 is the fractional concentration of vacancies and 𝑉𝑈.𝐶 is the volume of the unit cell. From this equation, a change in carrier concentration of 1 x 1018 cm-3 corresponds to a change of only 2.6 × 10 ―3 % in vacancies. This obviously is in conflict with the lead occupancy values refined from crystallographic experiments, which are three orders of magnitude larger. The temperature behavior of the different samples is also quite similar, with two distinct regions. At room temperature, the extrinsic region is clear, but around 623 K the behavior changes, and the charge carrier concentration increases. This is the region where the minority carriers are activated, and bipolar conduction is expected. If the carrier concentration was significantly different between the samples, this onset temperature of bipolar conduction would vary as well, but across all samples, the onset is approximately equal. All samples are n-type, but comparing with early articles, we would expect S-rich PbS to be p-type.41, 58 The fact that p-type is not obtained in any sample suggest that an S-rich environment was not achieved during synthesis, which further reduces the confidence in the lead occupancy results from PXRD. It appears that the small variations introduced in the synthesis were not sufficient to achieve lead vacancies, probably due to sulphur evaporation.


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Figure 4. Last cycle cooling curves of a) Hall carrier concentration, b) resistivity, c) mobility and d) Seebeck coefficient for the different PbS samples. The vertical line in a) indicates the onset of bipolar conduction. The line in c) is a guide to the eye with a T-5/2 slope to demonstrate the overall temperature behavior of the mobility.

The resistivity of the samples can be seen in Figure 4b and they are similar both in thermal behavior and values. The room temperature values vary between 5 – 10 m cm and all samples show similar behavior with increasing temperature. Initially, the samples show metallic behavior as the charge carrier concentration is constant, which changes approximately at the beginning of the intrinsic region. The exact position of the resistivity peak changes slightly across the different samples. No clear correlation with the formal stoichiometry can be seen,



and the changes are most likely due to small variations in mobility or onset of bipolar conduction. The mobility of the samples is shown in Figure 4c and the 50 ℃ values are listed in Table 2. Using Matthiessen’s rule, the dominating electronic scattering mechanisms can be estimated. Matthiessen’s rule state that the total reciprocal mobility is a sum of the independent, reciprocal 1

1

scattering mechanisms (𝜇 = ∑𝑖𝜇𝑖 ). The scattering mechanisms expected to be present in lead sulfide at elevated temperatures are acoustic phonon scattering, 𝜇𝑎𝑐 ∝ 𝑇 ―3/2, polar optical phonon scattering, 𝜇𝑝𝑜 ∝ 𝑇 ―1/2 and ionized impurity scattering, 𝜇𝑖𝑚 ∝ 𝑇3/2.1 Any scattering by dislocations, lattice impurities or neutral point defects is assumed to be negligible as previously shown to be the case in lead chalcogenides.65,66 The data exhibits a 𝑇 ―5/2 dependency, showing no signs of ionized impurity scattering as would have been expected in the case of VPb** or VPb* vacancies. In addition, there seems to be no significant changes between samples. The 𝑇 ―5/2 dependency has previously been explained by a combination of the acoustic phonon scattering and the temperature dependency of mobility and effective mass of carriers, 𝑚 ∗ .67 This is supported by the slight change in temperature dependency seen across the entire temperature range. The fact that no ionized impurity scattering is present further cements the observation that no significant changes in the vacancy concentration are present in these samples. The high temperature Seebeck coefficient is plotted in Figure 4d and the values obtained at 50 ℃ are listed in Table 2. All samples are n-type corroborating the Hall charge carrier concentration data. All samples have a nondegenerate temperature dependency of the Seebeck coefficient as previously seen for undoped PbS. The upturn temperature varies slightly between 573 – 673 and the maximum value also varies from 340 – 420 V/K. Comparing with the literature, this behavior matches well with behavior previously found for n-type lead sulfide.10,13,68 The Seebeck coefficient is the property that varies most across all the samples, both in the maximum value and the value at 50 ℃. In the single band parabolic model the


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Seebeck coefficient is proportional to m*n-2/3, and since the charge carrier concentration does not change significantly between samples, the change must come from other effects such as the carrier effective mass. The carrier effective mass was calculated using the single band Kane model67 (see Table 2) instead of the parabolic band model since it better describes the slightly non-parabolic nature of the band.69 In the literature, the carrier effective mass is reported to be between 𝑚 ∗ = 0.22 ― 0.38 𝑚𝑒, an interval that encapsulates the values for most of the samples in the present study.1 DFT calculations of the band structure show that the main contributor to the conduction band is lead.70 Vacancies on the lead site would result in a lower density of states near the conduction band edge, resulting in a more disperse band and therefore a lower effective mass. As is obvious from Table 2, the trend is opposite, since less stoichiometric samples have higher effective mass. Therefore, vacancies cannot account for the changes in effective mass. In summary, a large change in the lead vacancy concentration should affect both the thermal and electrical properties to a greater extent than the observed small variations documented here. The refined differences in lead occupancies are expected to give a much greater variation in the physical properties. In conclusion, the physical property measurements contradict the crystallographic results.

Rietveld refinements Refinement of the lead occupancy was introduced by Christensen et. al.25 in order to improve the values of the refined chalcogenide atomic displacement parameters (ADPs), since stoichiometric refinements gave unphysically low values. Free refinement of the lead site occupancy results in ADPs more in line with previous neutron diffraction data on mineralogical samples.63 Christensen et al. note that Pb-site occupancy was not refined for the mineralogical samples concluding that they must have been close to stoichiometry. Examining the agreement factors obtained by Christensen et al. for models with or without Pb-site occupancy refinement



they are quite similar with the full Pb occupancy actually being slightly lower. To evaluate the quality of the fits for the two models in the present case, we have calculated residual electron density plots (Fourier summation of Fobs – Fcalc), Figure 5. The models with or without lead occupancy refinement have similar agreement factors, but the corresponding residual densities are different. In the model with lead occupancy refinement, a greater residual is observed around the sulfur site in sharp contrast to the model without lead occupancy refinement. Even though the overall refinement residual values are similar, the Fourier difference plot shows that the model with lead refinement has more intense residual features. In combination with analysis of physical property measurements, this indicates that the model with lead occupancy is not actually fitting the lead occupancy, but instead modelling a different effect in the data.

Figure 5. Residual density for PbS-07 obtained from models with or without lead occupancy refinement. The electron density isosurface is set at 0.35 e/Å3.

Table 3. Rietveld refinement parameters of samples PbS-04, 06, 07 and 08 measured during three different beamtimes. wR(obs) and wRp are the weighted R-factors of structure factor and profile respectively. Two different Rietveld model have seen used on the data. The first section


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uses the model described by Christensen et al.,25 the second section uses an improved model using preferred orientation. Further data are listed in the supporting information. Sample

Model from Christensen et al. wR(obs)

wRp

(%)

(%)

Dec-16

1.46

3.40

Jul-17

1.59

Dec-17

Pb Occ.

Model using preferred orientation S(Uiso)

Pb(Ueq)

wR(obs)

wRp

(103 Å2)

(103 Å2)

(%)

(%)

0.941(4)

9.3(3)

6.0(1)

0.94

3.24

3.00

0.974(3)

9.1(3)

7.46(9)

1.48

4.41

5.78

0.780(5)

18.6(6)

7.99(18)

Dec-16

1.57

3.06

0.943(3)

9.0(3)

Jul-17

2.11

2.80

0.943(3)

Dec-17

1.88

2.87

Dec-16

2.06

Jul-17 Dec-17

Pb Occ.

S(Uiso)

Pb(Ueq)

March

(103 Å2)

(103 Å2)

parameter

1.006(5)

6.3(3)

5.97(10)

1.168(4)

2.97

1.000(4)

8.0(3)

7.44(9)

1.102(5)

2.22

4.12

1.004(6)

6.6(4)

8.21(13)

1.380(3)

5.9(1)

0.82

2.91

1.004(4)

6.2(3)

5.88(8)

1.163(3)

10.3(3)

6.88(7)

2.02

2.75

0.979(4)

8.5(3)

6.84(7)

1.122(4)

0.984(3)

8.9(3)

7.59(9)

1.84

2.73

1.012(5)

7.6(3)

7.57(9)

1.103(5)

3.43

0.915(3)

10.2(3)

6.3(1)

1.70

3.09

1.007(4)

5.9(3)

6.32(9)

1.208(3)

4.99

7.84

0.758(6)

18.8(7)

9.76(27)

2.52

5.87

0.995(8)

6.1(5)

9.80(20)

1.412(4)

2.05

2.93

0.944(3)

9.1(3)

6.88(9)

2.03

2.87

0.984(4)

7.2(3)

6.85(9)

1.128(4)

Dec-16

1.63

3.21

0.918(3)

8.8(3)

5.2(1)

1.04

2.93

0.998(4)

6.1(3)

6.11(9)

1.194(3)

Dec-17

1.92

2.62

0.973(3)

8.8(3)

6.92(8)

1.86

2.59

1.002(4)

7.4(3)

6.90(8)

1.106(4)

PbS𝟎𝟖

PbS𝟎𝟕

PbS𝟎𝟔

PbS𝟎𝟒

To determine what effect in the data the Pb Site occupancy is actually modelling and to develop at better Rietveld model for this effect, data were measured on all samples at several different times. The starting model used was the same as Christensen et al. and is referred to as “Model from Christensen et al.”. Selected refinement parameters can be seen in the first section of Table 3 (All refinement parameters are listed in supporting information Tables S3, S5 and S7). Using this model, significant differences are observed in both the refinement residuals, the



refined lead occupancies, and ADPs even on data recorded from the exact same sample, packed in different capillaries. One example of this is the PbS-08 sample, where refinements show significant differences from three different measurements. The Rietveld refinement using the initial model on data from December 2017 is shown in Figure 6, and the fit is clearly not satisfactory. Figure 6. Refinement of PbS-08_dec_18. Insert shows the first three peaks, which are all

labelled with their respective Miller indices.

The inset in Figure 6 shows the first three peaks (111, 002 and 202) and the blue difference line illustrates that the 111 and 202 reflections are overestimated in the model, while the 002 reflection is underestimated. In general, the {hhh} peaks are overestimated and the {00l} peaks are underestimated by the Rietveld model. This is traditionally interpreted as preferred orientation in the measured crystallites. A Rietveld model incorporating preferred orientation using the March-Dollase model48 was developed and employed to all data sets. Relevant refinement parameters can be seen in Table 3, section labeled “Model using preferred orientation” (All refinement parameters can be seen in supporting information, Tables S4, S6 and S8). The new model not only results in better agreement factors (R-values) but also reproduces other experimental values better. First, the sulfur ADPs are in the range 0.006-0.008


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Å-2 which is in line with values reported from single crystal71 and neutron63 experiments. Second, the Pb occupancy refines to 0.98-1.00, which is in line with the physical properties measurements. Therefore, we conclude that the model using preferred orientation is better for the lead sulfide data. It is worth noting that the new model still employs the Stephens model for refinement of anisotropic microstrain. Christensen et al. observed that {00l}-peaks are sharper than other peaks by observing the inequality 2S400 < S220. After employing the new model we still observe that 2S400 < S220 (see also supporting information). In a cubic structure 2S400 = S220, and the fact that 2S400 < S220 implies a local symmetry reduction. In order to probe if preferred orientation is an actual sample effect, TEM images were recorded on sample PbS-04, Figure 7, and no signs of preferred orientation are observed. The particle shapes appear random and most grains are polycrystalline with random orientation of the individual sections (insert of Figure 7). The experimental setup for the synchrotron PXRD measurements is designed to minimize preferred orientation since the samples are rotated during measurements, and highly homogenous powders are used in the capillaries (ethanol floating sample preparation technique, see Experimental Section). Furthermore, the PbS structure is cubic with no preferred growth direction. The preferred orientation parameters in the Rietveld model therefore must describe a different effect in the data, similar to what we concluded for the occupancy. To see if they model the same effect, the lead occupancy from the initial model is plotted against the preferred orientation parameter from the new model, Figure 8. A clear linear trend is seen between these two parameters and we conclude that they attempt to model the same effect in the data. In the following section, we will propose a new model that explains both the observed apparent preferred orientation and the inequality in microstrain. Further discussion of correlations in the refinement parameters can be found in Supporting Information VI.



Figure 7. TEM images of PbS-04. The sample was treated in the exact same way as before the packing into capillaries for PXRD measurements. All particles are polycrystalline and shows no sign on any preferred orientation. The zoom-in shows single-grain sections, indicated by the white arrows of the larger particle, note that the directions of the arrows are not correlated with the orientation of the grains. All these areas have randomly distributed crystallographic orientations.

Figure 8. Plot to show the correlation between refinement of the lead occupancy parameter and the preferred orientation.


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Reversability of parameters after heat treatment All PXRD measurements were done on powders from the direct synthesis, which were subsequently grinded and ball milled. This opens the question if the anomalies seen in the refinement parameters are remnants of the treatment of the powders and thereby from inhomogeneities between the measured samples and thermodynamically stable PbS. In Figure 3 a clear hysteresis is seen between the first heating and subsequent heating/cooling curves. This indicates that there is indeed some annealing during the first heating cycle. To examine if this affects the crystallographic parameters, powders of PbS-05 were heated from 300 K to 1000 K while measuring PXRD at 100 K intervals. A final measurement was done at 300 K after cooling. All refinement parameters from this can be seen in supporting information Table S1 (Without refinement of preferred orientation) and S2 (With refinement of preferred orientation). The most interesting parameters are the March parameter, the lead occupancy, the anisotropic strain and the unit cell parameter. The unit cell parameter decreases from 5.93162(3) Å to 5.92865(14) Å at 300 K after cycling. This could indicate that something changes in the sample, such as strain or occupancy. Indeed it is also observed that the anisotropic strain disappears on heating, but it returns upon cooling again. The absolute values of the strain has decreased during the thermal treatment and the ratio between S400 and S220 is now much closer to 2S400 = S220. This indicates that there is indeed induced strain from the powder treatment, but the fact that the strain returns upon cooling also indicates that there is intrinsic strain in the structure and that the strain refined in the samples is a mixture of the intrinsic and induced strain. The change in unit cell parameter could also explain the scattering of unit cell parameters found in Figure 2. Interestingly, the effect causing changes in the March parameter (and occupancy) does not seems to disappear, but only increase in strength after the thermal cycling, and thereby suggesting that they are indeed an intrinsic property in PbS.



Correlated disorder We suggest that direction-dependent correlated disorder discussed in recent literature28,72 can be the physical phenomena that explains both the observed anomalous peak broadening, as expressed through the inequality 2S400 < S220 from the anisotropic Stephens model, as well as the changes in observed intensities, which could be modelled through an apparent preferred orientation in the direction.

Figure 9. Sketch of the diffuse scattering in the HK0 plane observed for PbTe by Sangiorgio et al.28 The red shapes represent the observed diffuse scattering around the Bragg peaks. Peaks belonging the (H00) group have very strong diffuse scattering in a thin oblate in the KL plane of reciprocal space. Equivalently the 0K0 peaks have strong diffuse scattering in in the H and L directions. Other peaks, e.g. those in the (HH0) group have strong diffuse scattering extending in all three directions in reciprocal space. The thickness of the gray shells show the extent of broadening the diffuse scattering will have in a one-dimensional powder diffraction pattern. Thick black lines in the gray shells show the Bragg peak distance from the origin of reciprocal space.


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In PbTe it has been found that atoms have large displacements from their ideal positions in the average structure, but that locally there are strong correlations along the crystallographic axes tending to keep the interatomic distances close to their ideal values.28 These highly correlated atomic displacements along the crystallographic axes result in longer coherence lengths in those directions and shorter coherence lengths in other directions with the shortest coherence length along the direction of the space-diagonal. This disorder leads to diffuse scattering. The shorter coherence lengths along the space diagonal compared with the [001] directions leads to broad diffuse scattering around peaks along and a shaper diffuse scattering around peaks along . Figure 9 shows a sketch of the diffuse scattering for PbTe in the HK0 plane as reported by Sangiorgio et al.28 The red shapes show the regions of strong diffuse scattering around the Bragg peaks. (H00) peaks show strong diffuse scattering in the K and L directions, while they are sharp along H. Similarly the (0K0) peaks have strong diffuse scattering extending in the H and L directions. In contrast to this, other peaks such as (HH0) have diffuse scattering extending along all three directions of reciprocal space. When measuring the X-ray scattering from a powder, only the one-dimensional projection of the full three-dimensional scattering is collected. This means that all scattering with the same distance to the reciprocal space center (000) will be found at the same angle in the PXRD. As shown by the gray circles in Figure 9, the diffuse scattering around the (H00) peaks is found within a narrow shell around the Bragg peak, while the diffuse scattering around e.g. the (HH0) peaks is within a much wider shell. The results of this is that for powder diffraction data the (H00) peaks will have a too high intensity, as all the diffuse scattering is added to the Bragg peak in a very narrow angular range. In contrast, the (HH0) and other peaks will have a broader diffuse scattering contribution to the Bragg peak. In this case where the diffuse scattering has maxima on top of the Bragg peaks it is not possible for the Rietveld model to separate the



Page 26 of 39

diffuse from the Bragg scattering. While some of this broad diffuse scattering around the (HH0) peaks will be included as a background in the Rietveld model, some of it will also contribute to a peak broadening. For PbTe the effect of diffuse scattering on the powder diffraction data will be to give too high intensities to peaks while making peaks with a higher character too broad. It was reported by Bozin et al.24 that PbS has the same type of disorder as PbTe. It would therefore be expected that the diffuse scattering gives the same effects on the powder diffraction data for PbS, in excellent agreement with our observations. In the Rietveld refinement the effect of this diffuse scattering can be modelled through the anisotropic strain and preferred orientation parameters, as these have similar effects on the calculated diffraction pattern. The reflection-dependent broadening can be modeled as anisotropic micro-strain in the Rietveld refinement, using the Stephens model (strain causes peak broadening, similar to the diffuse scattering). The anisotropic model allows us to refine two different broadening parameters, S400 and S220. These parameters describe broadening of different peak classes. Our refinement shows the inequality 2S400 < S220, effectively showing that peaks are narrower than others. Christensen et al.25 also related this difference to a longer correlation length along the crystallographic axes. The changes in peak intensity induced by the diffuse scattering can be modelled through preferred orientation. The March-Dollase model for preferred orientation is a smoothly varying weight-function described as:48

(

)

1 𝑊𝑝𝑓(𝛼) = 𝑥 + (1 ― 𝑥) 𝑟2cos2 𝛼 + sin2 𝛼 𝑟

―

3 2

#(2)#

where 𝑥 is the fraction of non-textured sample (always set to zero in this experiment), 𝑟 is the March parameter which dictates the shape of 𝑊𝑝𝑓 as well as the strength of the preferred orientation, and 𝛼 is the angle between the specific peak hkl vector, and the preferred orientation


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vector, defined to be [001] in this experiment (𝛼 will be corrected by 𝜋/2 due to the capillary geometry). For all data reported here, the refinement always gives a March-parameter larger than unity resulting in the weight function increasing the intensity for reflections with decreasing angle 𝛼 to the (001) vector, i.e. reflections will have the largest increase in intensity. This is the same effect we expect from the diffuse scattering. If the preferred orientation was not used in the model, unphysical occupancies for Pb were instead obtained as discussed earlier. In the supporting information we further elaborate how the occupancy parameter has a similar effect on the calculated diffraction pattern. In summary, the strain and preferred orientation model is able to describe the effects of diffuse scattering from correlated disorder, here shown in the lead chalcogenide system. We conclude that both the strain and preferred orientation in the model are not necessarily real physical effects in the sample, but artefacts due to the effects of diffuse scattering on PXRD data.

Comparison with similar compounds The directional dependent long-range correlations have been proposed to stem from Pb lone pairs pointing along the axis directions, which mediates Pb-Te dimer formation along this axis. Christensen et al.25 argued that the presence of anomalous anisotropic strain is a consequence of an incipient cubic-to-orthorhombic phase transition seen in all lead chalcogenides. This is explained by hybridization between Pb(6s) and S(3p) orbitals, which is an extension of the sterically active lone-pair or resonant bonding theory.73 This hybridization was previously examined in PbS by ab initio calculations, and no signs of a sterically active lone pair was found.74 In addition to the cubic-to-orthorhombic phase transition at elevated pressures found for all lead chalcogenides, ab initio calculations have also predicted a cubic-to-rhombohedral transition for PbS at negative pressures (-6 GPa) from the NaCl structure to the α-GeTe crystal



structure.75 For PbS, the NaCl and α-GeTe structures are very close in energy, even at room temperature, whereas this is not the case for PbSe and PbTe. The cubic-to-rhombohedral transition is also seen in GeTe, where it changes from the NaCl structure to a R3m-structure (αGeTe) at high temperature.76,77 This cubic-to-rhombohedral transition can be seen as a distortion along the [111] axis resulting in a deviation from the  = 90o angle of the cubic structure. A recent study of the crystallography of this phase transition showed that there was a need to refine anisotropic strain parameters close to the transition temperature. In addition, the authors also report a change of the occupancy close to this phase transition, similar to the observations in the lead sulfide system.78 The expression of the sterically active lone pair through a greater s-p cation-anion hybridization indicates that there might be a strong link between the chemistry of PbS and GeTe. This is also supported by the behavior of chemical species of the same type. In the PbX (X = O, S, Se, Te) system, PbS, PbSe and PbTe are highly symmetric, while PbO has lower symmetry. In the GeX (X = O, S, Se, Te) only the GeTe is highly symmetric, while the others are lower symmetry. Thus, both GeTe and PbS are next to the lower symmetry species, and this might explain the similarity. The fact that both a cubic-to-orthorhombic and cubic-torhombohedral transition occur close to room temperature is an indication of a non-stable cubic phase, which in turn can explain the unusual models required to obtain a good Rietveld refinement fit. Overall, these observations correspond well with the recently proposed local dipole formation found in PbTe with correlated disorder along the directions.

CONCLUSIONS The structure and physical properties of PbS have been investigated on a range of samples synthesized with slight variation in the nominal lead content. The proposed lead vacancies found in previous articles are reproduced in Rietveld refinement, when using the same models


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as in previous studies. However, no signs of lead vacancies are observed in the physical properties of the samples leading to a clear contradiction between the structural and property analysis. A large discrepancy was found between different PXRD measurements on the exact same samples, and the March-Dollase model was introduced to describe preferred orientation effects while maintaining the large inequality in anisotropic strain parameters. Refinement of lead occupancies and preferred orientation have similar effect on the modelling of the present PXRD data on PbS. After the new model is introduced, there are no significant differences in stoichiometry between the different samples and there is full lead occupancy in all samples. The lead occupancy was simply a refinement parameter that models a different effect in the data. All studied samples were n-type suggesting that the synthesis was S poor instead of Pb poor likely due to S evaporation. We propose that the anisotropic strain and preferred orientation parameters describe the effects of diffuse scattering on the PXRD data due to direction dependent correlated disorder in the PbS structure. In other words, preferred orientation parameters together with anisotropic strain parameters are able to model the effects of diffuse scattering in this cubic system.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. ORCID Bo B. Iversen: 0000-0002-4632-1024 Notes The authors declare no competing financial interest



Page 30 of 39

ACKNOWLEDGEMENTS The author would like to thank Nils Lau Nyborg Broge for TEM measurements. The synchrotron radiation experiment at the SPring-8 synchrotron was conducted with the approval of the Japan Synchrotron Radiation Research Institute. The RIKEN SPring8 Center is thanked for access to the BL44B2 beamline. This research was supported by the Danish National Research Foundation (DNRF93). Affiliation with the Aarhus University Center for Integrated Materials Research (iMAT) is gratefully acknowledged.

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