Dopant Distributions in PbTe-Based ... - Semantic Scholar
Journal of ELECTRONIC MATERIALS, Vol. 41, No. 6, 2012
DOI: 10.1007/s11664-012-1972-2 Ó 2012 TMS
Dopant Distributions in PbTe-Based Thermoelectric Materials IVAN D. BLUM,1,5 DIETER ISHEIM,1,2 DAVID N. SEIDMAN,1,2,4 JIAQING HE,1 JOHN ANDROULAKIS,3 KANISHKA BISWAS,3 VINAYAK P. DRAVID,1 and MERCOURI G. KANATZIDIS3 1.—Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208-3108, USA. 2.—Northwestern University Center for Atom-Probe Tomography, Evanston, IL 60208-3108, USA. 3.—Department of Chemistry, Northwestern University, Evanston, IL 602083113, USA. 4.—e-mail: [email protected]. 5.—e-mail: [email protected]
Atom-probe tomography (APT) is utilized to characterize the dopant distribution in two thermoelectric materials systems: (1) PbTe-2 mol.%SrTe1 mol.%Na2Te, and (2) codoped PbTe-1.25 mol.%K-1.4 mol.%Na. We observe the presence of Na-rich precipitates of a few nanometers in diameter in both systems. Both concentration frequency distribution analyses and partial radial distribution functions are employed to analyze the tendency for dopant clustering detected by APT. In the codoped sample, K accumulates significantly in Na-rich precipitates, while in the Sr-containing sample, Sr is homogeneously distributed. High-resolution transmission electron microscopy also reveals the presence of precipitates having platelet morphology, which are oriented parallel to the {001} planes. Key words: PbTe, Na, K, precipitates, atom-probe tomography
INTRODUCTION PbTe systems are promising thermoelectric materials for electric power generation1 because they exhibit high figures of merit for nanostructured2 or doped materials.3,4 PbTe can be made p-type semiconducting by substitution of monovalent impurities on the lead sublattice. In principle, this can be achieved using alkali metals (Li, Na, K, Rb, Cs, Fr), Ag or Tl. Among these elements, Tl is extremely toxic, Ag is expensive and its price is currently increasing, and the ability of the elements with atomic radius larger than that of K to be efficient dopants remains an open question. Among the remaining elements, K and Na are usually assumed to form homogeneous solid solutions with PbTe at the small concentrations generally used for doping.4,5 A recent transmission electron microscopy (TEM) study has shown, however, that Na and K formed nanostructures in PbTe doped materials.6 In this research, we use atom-probe tomography (APT)7 to analyze the dopant distribution of K and Na in PbTe-based systems. APT is useful for characterization of thermoelectric materials.8–10 In (Received July 18, 2011; accepted January 22, 2012; published online February 24, 2012)
combination with TEM analyses, APT is utilized to characterize the dopant distributions in two systems: PbTe-2 mol.%SrTe-1 mol.%Na2Te, where Sr-rich precipitates are known to reduce the lattice thermal conductivity,11 and PbTe-1.25 mol.%K-1.4 mol.%Na, where K and Na codoping improves the power factor.4 EXPERIMENTAL PROCEDURES The first sample, with composition of PbTe2 mol.%SrTe doped with 1 mol.% Na2Te, was synthesized by mixing appropriate amounts of highpurity Pb, Te, Na2Te, and SrTe, with total mass of 10 g, in carbon-coated quartz tubes in an argon-filled glove box. The tubes were sealed under vacuum (104 Torr) and heated at 1323 K for 10 h. Next, the samples were slowly cooled to 873 K at a rate of 11 K h1 and then cooled to room temperature over 15 h. Other samples, with chemical formulas Pb0.9875yK0.0125NayTe (PbTe-1.25 mol.%K-Y mol.%Na, Y = 0.4, 0.8, or 1.4), were produced by reacting high-purity metals as starting materials. Approximately 15 mg K and 4.5 mg to 12 mg Na (per 10 g of product) were placed in carbon-coated silica tubes, together with the appropriate amounts of Pb and Te, inside a glove box under nitrogen atmosphere. The 1583
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tubes were heated at 1323 K for 4 h under vacuum (104 Torr), prior to cooling them to room temperature over a period of 3 h. The two PbTe-2 mol.%SrTe-1 mol.%Na2Te and PbTe-1.25 mol.%K-1.4 mol.%Na samples were prepared for x-ray diffraction studies by grinding them under Ar atmosphere to particle diameter of 30 lm. They were analyzed utilizing powder x-ray diffraction techniques employing synchrotron radiation ˚ ) while the sample was rocked around (k = 0.7106 A its position with a range of ±1.5° at each acquisition step. The same samples were also prepared for APT analyses using dual-beam focused ion-beam (FIB) microscopy.12 The APT analyses were performed using a local-electrode atom-probe (LEAP) tomograph (Cameca LEAP-4000X Si, Madison, WI), equipped with a picosecond ultraviolet (UV) laser (wavelength = 355 nm). During the analysis, the sample was maintained at 32.4 K, a laser energy of 30 pJ pulse1 was employed at pulse repetition rate of 250 kHz, utilizing a target evaporation rate of 0.005 atom pulse1. The PbTe-2 mol.%SrTe-1 mol.%Na2Te sample was also analyzed using voltage pulsing, with an evaporation rate of 0.005 atom pulse1, a pulse fraction (pulse voltage/steady-state dc voltage) of 15%, and a pulse repetition rate of 200 kHz. Individual ions were detected using a two-dimensional microchannel plate detector with detection efficiency of 50%. The data were analyzed using the program IVAS v 3.4.3 (Cameca, Madison, WI). During the APT analyses, low-density hkl poles and zone lines were observed, giving information about the crystallographic orientation of the sample.13 In the two analyses presented in this article, one hkl pole could be observed with zone lines having threefold symmetry, indicating it to be a (111) pole. The atoms were evaporated in different molecules: Na+, K+, Te+, Te2+, Te+2, + 2+ + + Te2+ 2 , Pb , Pb , PbTe , and PbTe2 . Among these ions, 2+ + Te and Te2 have similar mass-to-charge state ratios, and thus they have several overlapping peaks in the mass spectrum. The contribution of each ion to the overall composition was measured by deconvolution of the corresponding peaks using a fitting procedure. Thin sections of the PbTe-1.25 mol.%K-Y mol.%Na (Y = 0.4 or 0.8) specimens were examined utilizing a
JEOL 2100F TEM. The samples were cut into 3-mm-diameter discs using a disc cutter, then ground, dimpled, polished, and subsequently Ar+ion milled on a stage cooled with liquid nitrogen. RESULTS The two samples with nominal compositions PbTe2 mol.%SrTe-1 mol.%Na2Te and PbTe-1.25 mol.%K1.4 mol.%Na were analyzed using synchrotron x-ray diffraction (Fig. 1). All observed peaks correspond to the PbTe phase, which has the NaCl structure. Both samples were analyzed by APT and their bulk atomic compositions measured (Table I). The Pb and Te concentrations are relatively close to the nominal values for the two samples, while the Sr and Na concentrations are lower than the nominal values for the first sample as well as K for the second sample. Figure 2 displays the distributions of Na, Sr or K atoms in three-dimensional (3D) APT reconstructions of the two samples. The data presented in Fig. 2a and b are projections through 10-nm-thick slices, extracted from the total reconstructed volumes of 17 million atoms and 82 million atoms, respectively, and containing approximately 1 million detected atoms each. In both samples, Na-rich
Fig. 1. Powder x-ray diffraction spectrum for PbTe-2 mol.%SrTe˚ ). 1 mol.%Na2Te and PbTe-1.25 mol.%K-1.4 mol.%Na (k = 0.7106 A Only the diffraction peaks corresponding to PbTe (rock-salt structure) are observed. The ordinate scale is logarithmic.
Table I. Bulk elemental concentrations measured by atom-probe tomography PbTe-2 mol.%SrTe-1 mol.%Na2Te
Pb Te Na Sr
Nominal (at.%)
Measured (at.%)
48.3% 49.8% 1.0% 1.0%
48.5% 50.5% 0.56% 0.43%
PbTe-1.25 mol.%K-1.4 mol.%Na
Pb Te Na K
Nominal (at.%)
Measured (at.%)
48.7% 50.0% 0.7% 0.63%
45.6% 53.7% 0.60% 0.12%
The concentrations are given in atomic percent. The nominal atomic concentration for each element is calculated from the nominal molar concentration.
Dopant Distributions in PbTe-Based Thermoelectric Materials
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Fig. 2. Three-dimensional reconstruction of APT data for the samples: (a) PbTe-2 mol.%SrTe-1 mol.%Na2Te and (b) PbTe-1.25 mol.%K1.4 mol.%Na. The data presented are for 10-nm-thick slices extracted from the 3D reconstructed volumes. Na and Sr (or Na and K) atomic distributions are presented separately, and Pb and Te atoms are omitted for clarity. Some Na in homogeneities are encircled with black dashed lines. The indicated [111] directions are parallel to the plane of the extracted slices.
regions are observed. The largest Na-rich regions have flat morphologies, with thicknesses varying from 1 nm to 3 nm and diameters ranging from a few nanometers to 10 nm. These platelets were observed in different analyses, but they did not exhibit any consistent orientation with respect to the crystallographic orientation of the microtip. The distributions of Na, Sr, and K atoms are also analyzed by calculating a concentration frequency distribution for each sample (Fig. 3a, b). In this method, each reconstructed volume is divided into
blocks of an identical number of atoms and the concentration distribution of all the subvolumes (histogram) is compared with the binomial distribution (solid lines), which describes a random atomic distribution having a composition equal to the measured composition.14 Blocks of 100 atoms were used for Na and Sr, while blocks of 250 atoms were used for the K concentration frequency distribution because of its lower concentration. In both samples, the concentration distributions of Sr and K match the binomial distributions, within the
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Fig. 4. Partial radial distribution functions (RDFs) with respect to Na atoms in the samples: (a) PbTe-2 mol.%SrTe-1 mol.%Na2Te and (b) PbTe-1.25 mol.%K-1.4 mol.%Na. The concentrations are normalized to the measured bulk concentrations: a value greater than unity indicates clustering.
Fig. 3. Concentration frequency distributions for (a) PbTe2 mol.%SrTe-1 mol.%Na2Te and (b) PbTe-1.25 mol.%K-1.4 mol.%Na. The 3D APT reconstructed volume is divided into blocks of identical numbers of atoms. The concentration frequency distributions (histograms) of the blocks are compared, for each element, with the calculated binomial distribution (solid lines) representing a random distribution with the same composition. Blocks of 100 atoms are used for Na and Sr, and blocks of 250 atoms are used for K concentration frequency distributions. The error bars correspond to ±2r calculated from counting statistics.
statistical limit of error. The Na distribution confirms the presence of nonrandom Na-rich regions for the codoped sample (Fig. 3b) and also, but to a smaller degree, for the Sr-containing sample (Fig. 3a). This method is, however, only sensitive to precipitates that are large enough to affect the composition of at least one block. To analyze atomic distributions at a smaller length scale, the partial radial distribution function (RDF) with respect to Na atoms was calculated (Fig. 4). The concentration represented in the partial RDF is normalized to the concentration measured in the reconstructed volume. Thus, a value greater than unity for an element implies that Na has, on average, more neighbors of that element than predicted for a random distribution. In the cases where APT achieves spatial resolution better than the atomic nextneighbor interatomic distance, the partial RDF can be used to determine the chemical nature of the nearest neighbors (NNs), thereby allowing measurement of ordering phenomena.15 When the spatial resolution does not resolve atomic next-neighbor positions, this method is still sensitive to clustering or precipitation phenomena on longer length scales. For the PbTe-2 mol.%SrTe-1 mol.%Na2Te sample (Fig. 4a), this analysis confirms that Na is inhomogeneously distributed at the nanometer scale, while
Sr is almost homogeneously distributed. Indeed, the highest value achieved by the partial RDF of Sr is only 1.07 ± 0.05 times the concentration of Sr of the bulk. The maximum of the partial Na-Na RDF is 1.66 ± 0.06 and the width at half-maximum is 1.2 nm, which is an estimate of the average radius of the precipitates. A similar sample was also analyzed by APT using voltage pulsing (Fig. 4a, dashed line). In this volume, Na is also inhomogeneously distributed, but less precipitation is observed (i.e., fewer precipitates and/or precipitates having lower Na concentration), as illustrated by the smaller values for the partial Na-Na RDF: 1.20 ± 0.03. For the second system, PbTe-1.25 mol.%K-1.4 mol.%Na, we observe also Na inhomogeneities and to a lesser extent for K with maximum partial RDF values of 1.66 ± 0.02 and 1.11 ± 0.04 and full-widths at halfmaximum of 1 nm and 0.8 nm, respectively. A curve similar to the partial Na-K RDF was observed for K-Na and K-K partial RDFs (not displayed), indicating that both Na and K form precipitates and can coexist, on average, in the same precipitates. Small variations in the partial RDFs of Pb and Te with respect to Na or K were also observed (not shown here), corresponding to changes in concentrations
DOI: 10.1007/s11664-012-1972-2 Ó 2012 TMS
Dopant Distributions in PbTe-Based Thermoelectric Materials IVAN D. BLUM,1,5 DIETER ISHEIM,1,2 DAVID N. SEIDMAN,1,2,4 JIAQING HE,1 JOHN ANDROULAKIS,3 KANISHKA BISWAS,3 VINAYAK P. DRAVID,1 and MERCOURI G. KANATZIDIS3 1.—Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208-3108, USA. 2.—Northwestern University Center for Atom-Probe Tomography, Evanston, IL 60208-3108, USA. 3.—Department of Chemistry, Northwestern University, Evanston, IL 602083113, USA. 4.—e-mail: [email protected]. 5.—e-mail: [email protected]
Atom-probe tomography (APT) is utilized to characterize the dopant distribution in two thermoelectric materials systems: (1) PbTe-2 mol.%SrTe1 mol.%Na2Te, and (2) codoped PbTe-1.25 mol.%K-1.4 mol.%Na. We observe the presence of Na-rich precipitates of a few nanometers in diameter in both systems. Both concentration frequency distribution analyses and partial radial distribution functions are employed to analyze the tendency for dopant clustering detected by APT. In the codoped sample, K accumulates significantly in Na-rich precipitates, while in the Sr-containing sample, Sr is homogeneously distributed. High-resolution transmission electron microscopy also reveals the presence of precipitates having platelet morphology, which are oriented parallel to the {001} planes. Key words: PbTe, Na, K, precipitates, atom-probe tomography
INTRODUCTION PbTe systems are promising thermoelectric materials for electric power generation1 because they exhibit high figures of merit for nanostructured2 or doped materials.3,4 PbTe can be made p-type semiconducting by substitution of monovalent impurities on the lead sublattice. In principle, this can be achieved using alkali metals (Li, Na, K, Rb, Cs, Fr), Ag or Tl. Among these elements, Tl is extremely toxic, Ag is expensive and its price is currently increasing, and the ability of the elements with atomic radius larger than that of K to be efficient dopants remains an open question. Among the remaining elements, K and Na are usually assumed to form homogeneous solid solutions with PbTe at the small concentrations generally used for doping.4,5 A recent transmission electron microscopy (TEM) study has shown, however, that Na and K formed nanostructures in PbTe doped materials.6 In this research, we use atom-probe tomography (APT)7 to analyze the dopant distribution of K and Na in PbTe-based systems. APT is useful for characterization of thermoelectric materials.8–10 In (Received July 18, 2011; accepted January 22, 2012; published online February 24, 2012)
combination with TEM analyses, APT is utilized to characterize the dopant distributions in two systems: PbTe-2 mol.%SrTe-1 mol.%Na2Te, where Sr-rich precipitates are known to reduce the lattice thermal conductivity,11 and PbTe-1.25 mol.%K-1.4 mol.%Na, where K and Na codoping improves the power factor.4 EXPERIMENTAL PROCEDURES The first sample, with composition of PbTe2 mol.%SrTe doped with 1 mol.% Na2Te, was synthesized by mixing appropriate amounts of highpurity Pb, Te, Na2Te, and SrTe, with total mass of 10 g, in carbon-coated quartz tubes in an argon-filled glove box. The tubes were sealed under vacuum (104 Torr) and heated at 1323 K for 10 h. Next, the samples were slowly cooled to 873 K at a rate of 11 K h1 and then cooled to room temperature over 15 h. Other samples, with chemical formulas Pb0.9875yK0.0125NayTe (PbTe-1.25 mol.%K-Y mol.%Na, Y = 0.4, 0.8, or 1.4), were produced by reacting high-purity metals as starting materials. Approximately 15 mg K and 4.5 mg to 12 mg Na (per 10 g of product) were placed in carbon-coated silica tubes, together with the appropriate amounts of Pb and Te, inside a glove box under nitrogen atmosphere. The 1583
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tubes were heated at 1323 K for 4 h under vacuum (104 Torr), prior to cooling them to room temperature over a period of 3 h. The two PbTe-2 mol.%SrTe-1 mol.%Na2Te and PbTe-1.25 mol.%K-1.4 mol.%Na samples were prepared for x-ray diffraction studies by grinding them under Ar atmosphere to particle diameter of 30 lm. They were analyzed utilizing powder x-ray diffraction techniques employing synchrotron radiation ˚ ) while the sample was rocked around (k = 0.7106 A its position with a range of ±1.5° at each acquisition step. The same samples were also prepared for APT analyses using dual-beam focused ion-beam (FIB) microscopy.12 The APT analyses were performed using a local-electrode atom-probe (LEAP) tomograph (Cameca LEAP-4000X Si, Madison, WI), equipped with a picosecond ultraviolet (UV) laser (wavelength = 355 nm). During the analysis, the sample was maintained at 32.4 K, a laser energy of 30 pJ pulse1 was employed at pulse repetition rate of 250 kHz, utilizing a target evaporation rate of 0.005 atom pulse1. The PbTe-2 mol.%SrTe-1 mol.%Na2Te sample was also analyzed using voltage pulsing, with an evaporation rate of 0.005 atom pulse1, a pulse fraction (pulse voltage/steady-state dc voltage) of 15%, and a pulse repetition rate of 200 kHz. Individual ions were detected using a two-dimensional microchannel plate detector with detection efficiency of 50%. The data were analyzed using the program IVAS v 3.4.3 (Cameca, Madison, WI). During the APT analyses, low-density hkl poles and zone lines were observed, giving information about the crystallographic orientation of the sample.13 In the two analyses presented in this article, one hkl pole could be observed with zone lines having threefold symmetry, indicating it to be a (111) pole. The atoms were evaporated in different molecules: Na+, K+, Te+, Te2+, Te+2, + 2+ + + Te2+ 2 , Pb , Pb , PbTe , and PbTe2 . Among these ions, 2+ + Te and Te2 have similar mass-to-charge state ratios, and thus they have several overlapping peaks in the mass spectrum. The contribution of each ion to the overall composition was measured by deconvolution of the corresponding peaks using a fitting procedure. Thin sections of the PbTe-1.25 mol.%K-Y mol.%Na (Y = 0.4 or 0.8) specimens were examined utilizing a
JEOL 2100F TEM. The samples were cut into 3-mm-diameter discs using a disc cutter, then ground, dimpled, polished, and subsequently Ar+ion milled on a stage cooled with liquid nitrogen. RESULTS The two samples with nominal compositions PbTe2 mol.%SrTe-1 mol.%Na2Te and PbTe-1.25 mol.%K1.4 mol.%Na were analyzed using synchrotron x-ray diffraction (Fig. 1). All observed peaks correspond to the PbTe phase, which has the NaCl structure. Both samples were analyzed by APT and their bulk atomic compositions measured (Table I). The Pb and Te concentrations are relatively close to the nominal values for the two samples, while the Sr and Na concentrations are lower than the nominal values for the first sample as well as K for the second sample. Figure 2 displays the distributions of Na, Sr or K atoms in three-dimensional (3D) APT reconstructions of the two samples. The data presented in Fig. 2a and b are projections through 10-nm-thick slices, extracted from the total reconstructed volumes of 17 million atoms and 82 million atoms, respectively, and containing approximately 1 million detected atoms each. In both samples, Na-rich
Fig. 1. Powder x-ray diffraction spectrum for PbTe-2 mol.%SrTe˚ ). 1 mol.%Na2Te and PbTe-1.25 mol.%K-1.4 mol.%Na (k = 0.7106 A Only the diffraction peaks corresponding to PbTe (rock-salt structure) are observed. The ordinate scale is logarithmic.
Table I. Bulk elemental concentrations measured by atom-probe tomography PbTe-2 mol.%SrTe-1 mol.%Na2Te
Pb Te Na Sr
Nominal (at.%)
Measured (at.%)
48.3% 49.8% 1.0% 1.0%
48.5% 50.5% 0.56% 0.43%
PbTe-1.25 mol.%K-1.4 mol.%Na
Pb Te Na K
Nominal (at.%)
Measured (at.%)
48.7% 50.0% 0.7% 0.63%
45.6% 53.7% 0.60% 0.12%
The concentrations are given in atomic percent. The nominal atomic concentration for each element is calculated from the nominal molar concentration.
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Fig. 2. Three-dimensional reconstruction of APT data for the samples: (a) PbTe-2 mol.%SrTe-1 mol.%Na2Te and (b) PbTe-1.25 mol.%K1.4 mol.%Na. The data presented are for 10-nm-thick slices extracted from the 3D reconstructed volumes. Na and Sr (or Na and K) atomic distributions are presented separately, and Pb and Te atoms are omitted for clarity. Some Na in homogeneities are encircled with black dashed lines. The indicated [111] directions are parallel to the plane of the extracted slices.
regions are observed. The largest Na-rich regions have flat morphologies, with thicknesses varying from 1 nm to 3 nm and diameters ranging from a few nanometers to 10 nm. These platelets were observed in different analyses, but they did not exhibit any consistent orientation with respect to the crystallographic orientation of the microtip. The distributions of Na, Sr, and K atoms are also analyzed by calculating a concentration frequency distribution for each sample (Fig. 3a, b). In this method, each reconstructed volume is divided into
blocks of an identical number of atoms and the concentration distribution of all the subvolumes (histogram) is compared with the binomial distribution (solid lines), which describes a random atomic distribution having a composition equal to the measured composition.14 Blocks of 100 atoms were used for Na and Sr, while blocks of 250 atoms were used for the K concentration frequency distribution because of its lower concentration. In both samples, the concentration distributions of Sr and K match the binomial distributions, within the
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Fig. 4. Partial radial distribution functions (RDFs) with respect to Na atoms in the samples: (a) PbTe-2 mol.%SrTe-1 mol.%Na2Te and (b) PbTe-1.25 mol.%K-1.4 mol.%Na. The concentrations are normalized to the measured bulk concentrations: a value greater than unity indicates clustering.
Fig. 3. Concentration frequency distributions for (a) PbTe2 mol.%SrTe-1 mol.%Na2Te and (b) PbTe-1.25 mol.%K-1.4 mol.%Na. The 3D APT reconstructed volume is divided into blocks of identical numbers of atoms. The concentration frequency distributions (histograms) of the blocks are compared, for each element, with the calculated binomial distribution (solid lines) representing a random distribution with the same composition. Blocks of 100 atoms are used for Na and Sr, and blocks of 250 atoms are used for K concentration frequency distributions. The error bars correspond to ±2r calculated from counting statistics.
statistical limit of error. The Na distribution confirms the presence of nonrandom Na-rich regions for the codoped sample (Fig. 3b) and also, but to a smaller degree, for the Sr-containing sample (Fig. 3a). This method is, however, only sensitive to precipitates that are large enough to affect the composition of at least one block. To analyze atomic distributions at a smaller length scale, the partial radial distribution function (RDF) with respect to Na atoms was calculated (Fig. 4). The concentration represented in the partial RDF is normalized to the concentration measured in the reconstructed volume. Thus, a value greater than unity for an element implies that Na has, on average, more neighbors of that element than predicted for a random distribution. In the cases where APT achieves spatial resolution better than the atomic nextneighbor interatomic distance, the partial RDF can be used to determine the chemical nature of the nearest neighbors (NNs), thereby allowing measurement of ordering phenomena.15 When the spatial resolution does not resolve atomic next-neighbor positions, this method is still sensitive to clustering or precipitation phenomena on longer length scales. For the PbTe-2 mol.%SrTe-1 mol.%Na2Te sample (Fig. 4a), this analysis confirms that Na is inhomogeneously distributed at the nanometer scale, while
Sr is almost homogeneously distributed. Indeed, the highest value achieved by the partial RDF of Sr is only 1.07 ± 0.05 times the concentration of Sr of the bulk. The maximum of the partial Na-Na RDF is 1.66 ± 0.06 and the width at half-maximum is 1.2 nm, which is an estimate of the average radius of the precipitates. A similar sample was also analyzed by APT using voltage pulsing (Fig. 4a, dashed line). In this volume, Na is also inhomogeneously distributed, but less precipitation is observed (i.e., fewer precipitates and/or precipitates having lower Na concentration), as illustrated by the smaller values for the partial Na-Na RDF: 1.20 ± 0.03. For the second system, PbTe-1.25 mol.%K-1.4 mol.%Na, we observe also Na inhomogeneities and to a lesser extent for K with maximum partial RDF values of 1.66 ± 0.02 and 1.11 ± 0.04 and full-widths at halfmaximum of 1 nm and 0.8 nm, respectively. A curve similar to the partial Na-K RDF was observed for K-Na and K-K partial RDFs (not displayed), indicating that both Na and K form precipitates and can coexist, on average, in the same precipitates. Small variations in the partial RDFs of Pb and Te with respect to Na or K were also observed (not shown here), corresponding to changes in concentrations
