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Abstract
The Seebeck coefficient is one of the key quantities of thermoelectric materials and routinely measured in various laboratories. There are, however, several ways to calculate the Seebeck coefficient from the raw measurement data. We compare these different ways to extract the Seebeck coefficient, evaluate the accuracy of the results, and show methods to increase this accuracy. We furthermore point out experimental and data analysis parameters that can be used to evaluate the trustworthiness of the obtained result. The shown analysis can be used to find and minimize errors in the Seebeck coefficient measurement and therefore increase the reliability of the measured material properties.
G. Karpinski is acknowledged for setting up the measurement routines, C. Stiewe and P. Ziolkowski for fruitful discussions about the paper. Financial support from the BMBF project INTEG (Grant No. 03X3555A) is gratefully acknowledged.
I. INTRODUCTION AND MOTIVATION
II. EXEMPLARY SETUP
III. FROM RAW MEASUREMENT DATA TO THE SEEBECK COEFFICIENT
IV. SIMULTANEOUS MEASUREMENTS
V. CONSISTENCY CHECKS AND ERROR INDICATIONS
VI. SUMMARY
Key Topics
 Thermocouples
 33.0
 Temperature measurement
 25.0
 Electric measurements
 24.0
 Interpolation
 17.0
 Contact resistance
 11.0
Figures
Sample holder for the measurement of the Seebeck coefficient. The sample is sandwiched between the holder and a support. The temperature gradient along the sample is established using one of the gradient heaters. The resulting thermoelectric voltages and temperatures are recorded using two thermocouples. The whole sample holder is situated in an oven; this allows for temperature dependent measurements.
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Sample holder for the measurement of the Seebeck coefficient. The sample is sandwiched between the holder and a support. The temperature gradient along the sample is established using one of the gradient heaters. The resulting thermoelectric voltages and temperatures are recorded using two thermocouples. The whole sample holder is situated in an oven; this allows for temperature dependent measurements.
Wiring scheme of the employed measurement setup. The two thermocouples are connected directly to the switch card and are used to obtain the temperatures T 1 and T 2. Furthermore, the voltage Uneg across sample and negative legs of the thermocouples as well as the voltage Upos across sample and positive legs are measured on the switch card; these are used to calculate the sample's Seebeck coefficient.
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Wiring scheme of the employed measurement setup. The two thermocouples are connected directly to the switch card and are used to obtain the temperatures T 1 and T 2. Furthermore, the voltage Uneg across sample and negative legs of the thermocouples as well as the voltage Upos across sample and positive legs are measured on the switch card; these are used to calculate the sample's Seebeck coefficient.
Measurement routine for the determination of the sample's Seebeck coefficient. The figure shows the time resolved profiles of the temperatures (top) as well as the voltages during one measurement cycle. For later analysis, only the values recorded after the heaters are switched off are used.
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Measurement routine for the determination of the sample's Seebeck coefficient. The figure shows the time resolved profiles of the temperatures (top) as well as the voltages during one measurement cycle. For later analysis, only the values recorded after the heaters are switched off are used.
Seebeck coefficient as obtained from Eq. (4) for the data shown in Figure 3 . The result is not independent of ΔT as would principally be expected. The reason for the behavior is a combination of offsets between the thermocouples and spurious voltages from within the measurement system. The effects of these offsets can be expressed in a simple model (using Eq. (7) ) and the result fits the experimental data reasonably well.
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Seebeck coefficient as obtained from Eq. (4) for the data shown in Figure 3 . The result is not independent of ΔT as would principally be expected. The reason for the behavior is a combination of offsets between the thermocouples and spurious voltages from within the measurement system. The effects of these offsets can be expressed in a simple model (using Eq. (7) ) and the result fits the experimental data reasonably well.
(a) Seebeck voltages U pos and U neg vs. temperature difference ΔT and (b) U pos vs. U neg . All three data sets show a linear behavior as indicated by the good agreement between measurement data and the corresponding linear fits. Also indicated are the results for the Seebeck coefficient (Eqs. (8)–(10) ), which agree with each other within 0.1%. The linear correlation coefficient R of the fit gives an indication about the agreement between raw data and linear fit; here R is very close to unity for all three data sets.
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(a) Seebeck voltages U pos and U neg vs. temperature difference ΔT and (b) U pos vs. U neg . All three data sets show a linear behavior as indicated by the good agreement between measurement data and the corresponding linear fits. Also indicated are the results for the Seebeck coefficient (Eqs. (8)–(10) ), which agree with each other within 0.1%. The linear correlation coefficient R of the fit gives an indication about the agreement between raw data and linear fit; here R is very close to unity for all three data sets.
Seebeck coefficient over temperature for a skutterudite sample. The plot compares the results of Eqs. (8)–(10) which basically lie on top of each other. The relative difference between S D , S E , and S F is smaller than 0.6% for the whole temperature range.
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Seebeck coefficient over temperature for a skutterudite sample. The plot compares the results of Eqs. (8)–(10) which basically lie on top of each other. The relative difference between S D , S E , and S F is smaller than 0.6% for the whole temperature range.
Linear correlation coefficient for the results shown in Figure 6 . The linear correlation coefficients and thus the quality of the fits used to obtain the Seebeck coefficient are approximately independent of temperature and close to unity over the whole temperature range.
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Linear correlation coefficient for the results shown in Figure 6 . The linear correlation coefficients and thus the quality of the fits used to obtain the Seebeck coefficient are approximately independent of temperature and close to unity over the whole temperature range.
Temperature of the six internal cold side temperature sensors of the employed switching card in the multimeter vs. measurement temperature and time. All temperatures are similar and vary only slowly over the course of the measurement. For the voltage to temperature conversion the temperature of the sensor closest to the respective thermocouple measurement channel is employed.
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Temperature of the six internal cold side temperature sensors of the employed switching card in the multimeter vs. measurement temperature and time. All temperatures are similar and vary only slowly over the course of the measurement. For the voltage to temperature conversion the temperature of the sensor closest to the respective thermocouple measurement channel is employed.
(a) Seebeck coefficients calculated from (8)–(10) . In contrast to the results shown in Figure 6 in this case the internal voltage to temperature conversion of the multimeter has been used to obtain the employed temperature differences. While all three curves follow the same trend it can be seen that S D and S E show some scattering and partially deviate significantly from S F . The linear correlation coefficients of the underlying linear fits in (b) show that the fits are better for S F .
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(a) Seebeck coefficients calculated from (8)–(10) . In contrast to the results shown in Figure 6 in this case the internal voltage to temperature conversion of the multimeter has been used to obtain the employed temperature differences. While all three curves follow the same trend it can be seen that S D and S E show some scattering and partially deviate significantly from S F . The linear correlation coefficients of the underlying linear fits in (b) show that the fits are better for S F .
U pos vs. U neg with and without temporal interpolation. For the raw data each measurement of U neg is related to the subsequent measurement of U pos . For the interpolated data the value is calculated from the temporal interpolation of the measurement data such that there is temporal match between U neg and U pos . Although the effect appears to be small and is hardly visible in the overview plot, the zoom in shows that the interpolation to the same time changes the values of U pos and thus the calculated Seebeck coefficient.
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U pos vs. U neg with and without temporal interpolation. For the raw data each measurement of U neg is related to the subsequent measurement of U pos . For the interpolated data the value is calculated from the temporal interpolation of the measurement data such that there is temporal match between U neg and U pos . Although the effect appears to be small and is hardly visible in the overview plot, the zoom in shows that the interpolation to the same time changes the values of U pos and thus the calculated Seebeck coefficient.
Seebeck coefficient over temperature obtained from raw measurement data and interpolated data. Here the relative error is about 2%, depending on sample properties and geometry it can vary between 1% and 5%.
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Seebeck coefficient over temperature obtained from raw measurement data and interpolated data. Here the relative error is about 2%, depending on sample properties and geometry it can vary between 1% and 5%.
Offset of the fit U pos vs. U neg with respect to measurement temperature. The offset is shown for two samples with different Seebeck coefficients; nevertheless resulting in similar offsets.
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Offset of the fit U pos vs. U neg with respect to measurement temperature. The offset is shown for two samples with different Seebeck coefficients; nevertheless resulting in similar offsets.
(a) Seebeck coefficient of a Zn4Sb3 sample over temperature. “α + β” corresponds to the Seebeck coefficient that is obtained if the whole data from the measurement routine as shown in Figure 3 are employed. The result if only the first half of the data are employed is labeled “α,” while the result from the second half is labeled “β.” For stable samples the three values should be very close to each other, but here it can be seen that between 420 K and 470 K “α” and “β” differ from each other. The reason is a change of the sample properties during the measurement, i.e., the sample has (on average) a different Seebeck coefficient during measurement “α” than at “β” at a given temperature. This is reflected in the correlation coefficient (b) that deviates stronger from unity in the region where the sample is thermally unstable. This shows that the linear correlation coefficient can be used to detect such changes.
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(a) Seebeck coefficient of a Zn4Sb3 sample over temperature. “α + β” corresponds to the Seebeck coefficient that is obtained if the whole data from the measurement routine as shown in Figure 3 are employed. The result if only the first half of the data are employed is labeled “α,” while the result from the second half is labeled “β.” For stable samples the three values should be very close to each other, but here it can be seen that between 420 K and 470 K “α” and “β” differ from each other. The reason is a change of the sample properties during the measurement, i.e., the sample has (on average) a different Seebeck coefficient during measurement “α” than at “β” at a given temperature. This is reflected in the correlation coefficient (b) that deviates stronger from unity in the region where the sample is thermally unstable. This shows that the linear correlation coefficient can be used to detect such changes.
Exemplary twopoint resistances of the two thermocouple channels and the two circuits used to measure U neg and U pos over temperature. Drastic changes in R(U neg ) or R(U pos ) can indicate poor contact between the sample and the thermocouples while comparison of R(T1) and R(T2) between different measurements can reveal changes of the thermocouples, e.g., by contamination.
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Exemplary twopoint resistances of the two thermocouple channels and the two circuits used to measure U neg and U pos over temperature. Drastic changes in R(U neg ) or R(U pos ) can indicate poor contact between the sample and the thermocouples while comparison of R(T1) and R(T2) between different measurements can reveal changes of the thermocouples, e.g., by contamination.
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Abstract
The Seebeck coefficient is one of the key quantities of thermoelectric materials and routinely measured in various laboratories. There are, however, several ways to calculate the Seebeck coefficient from the raw measurement data. We compare these different ways to extract the Seebeck coefficient, evaluate the accuracy of the results, and show methods to increase this accuracy. We furthermore point out experimental and data analysis parameters that can be used to evaluate the trustworthiness of the obtained result. The shown analysis can be used to find and minimize errors in the Seebeck coefficient measurement and therefore increase the reliability of the measured material properties.
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