The Seebeck coefficient is the most widely measured property specific to thermoelectric materials. The absolute Seebeck coefficient *S* determined from the Thomson effect is highly sensitive to systematic errors incurred in the determination of the material thermal conductivity and geometry and heat loss from the sample to surroundings caused by temperature differences. Here, we report a technique for the precision measurement of *S* based on the Thomson effect using an ac-dc technique. This technique utilizes accurate equivalent-amplitude ac and dc currents, which can eliminate the need for acquiring accurate thermal conductivity and geometry values. These parameters can be replaced by the precisely and readily measurable parameters of electrical resistance and temperature changes caused by the Joule effect. The correction term of the heat loss owing to heat transfer via the thermocouple vanishes upon calculating the ratio of the measured temperature changes for both ac and dc excitations. We obtain an *S* value of -4.8 μV/K ± 0.2 μV/K at a temperature of 300 K for platinum, which is most widely used as a reference, with an expanded relative uncertainty of 4% (2σ). The obtained *S* value of Pt is closely consistent with that obtained from the conventional method using the Thomson effect within the uncertainty, and importantly, the measurement uncertainty improves to an acceptable level, which is four times more precise.

## I. INTRODUCTION

The Seebeck coefficient is an essential indicator of the potential performance of thermoelectric materials used for thermoelectric power generators.^{1} The results of a recent international round-robin study of thermoelectric materials suggest that it is important to ensure absolute data accuracy when comparing the evaluated values acquired by several different laboratories.^{2} The determination of the Seebeck coefficient is particularly challenging because it is obtained by means of inherently relative measurements. The Seebeck coefficient of a thermoelectric material, *S*_{sample}, is evaluated via measuring the output of a thermocouple circuit,^{3} and is expressed as

where *S*_{reference} denotes the Seebeck coefficient of the reference material and Δ*V* the voltage induced by temperature difference Δ*T* across the sample. Therefore, the Seebeck coefficient of reference materials such as Pb, Pt, Cu, and W^{4–8} must be determined through separate experiments for the entire temperature range of interest. Two techniques are commonly used to measure the absolute Seebeck coefficient: at low temperatures, a superconducting material is used as the reference because the Seebeck coefficient is zero in the Meissner state. The main drawback of this method is that the highest measurable temperature is currently limited by the transition temperature *T*_{c} of the reference material, which is approximately 90 K for YBa_{2}Cu_{3}O_{7-x} (YBCO).^{9} At higher temperatures, the absolute Seebeck coefficient can be calculated by measuring the Thomson coefficient *μ* directly and using the Kelvin relation for the Seebeck coefficient:^{10,11}

where *T*_{1} and *T*_{2} represent the base temperature utilized when determining the first term *S*(*T*_{1}) using the superconductor reference material (which may be below *T*_{c}) and the base temperature of interest, respectively. Thus, it is obvious that the direct measurement of the Thomson coefficient can be utilized to determine the absolute Seebeck coefficient. However, it is difficult to apply this Thomson-effect-based scheme to accurately determine the Seebeck coefficient, because the conventional formula of the Thomson coefficient determined using only dc current requires accurate knowledge of the sample thermal conductivity and geometry,^{12} and further, it is sensitive to the heat loss caused by heat transfer via the thermocouple connected to the sample.^{6}

To address these technical issues, ac measurement techniques have been recently introduced to evaluate the Thomson coefficient: ac-dc techniques, which utilize accurate equivalent-amplitude ac and dc currents to measure the Thomson coefficient regardless of the thermal conductivity and geometry values,^{13} and the 2*ω* method, which is a phase-sensitive ac current technique using the temperature-dependence data of the sample resistivity.^{14} The ac-dc technique allows us to perform a direct temperature measurement due to the Thomson effect by using the thermocouple attached at the middle of the sample. Meanwhile, it requires sensitive dc voltage measurements as small as 100 nV from the thermocouple. Consequently, in this study wherein we use the ac-dc technique, we developed a cryostat that enabled us to control the temperature in a very stable manner and thus perform sensitive measurements relative to the case of our previous room-temperature apparatus. Furthermore, we investigated the effects of heat loss from the sample and displacement of the thermocouple from the middle of the sample on the measurement of *μ* and conducted a proof-of-principle experiment to precisely measure the *S* value of platinum, which is the most widely used reference, by utilizing our ac-dc technique. Here, we present our heat transfer analysis, the precise measurement results of *μ* and *S* of platinum in the range from 90 K to 300 K, as well as the uncertainty analysis.

## II. MEASUREMENT PRINCIPLE

Figure 1(a) shows the schematic graph of the temperature change caused by the Thomson effect when ac and dc currents are applied alternately to the sample. In the case of ac current application, Thomson heating is canceled out when the ac frequency is sufficiently high, because the degree of Thomson effect heating or absorption depends on the polarity of the currents; thus, the temperature rise is solely due to the Joule effect. In contrast, in the dc case, the temperature rise can be caused by both the Joule and Thomson effects. Let us consider the heat conduction model under a temperature difference and the flow of equivalent dc and ac current as per the schematic shown in Fig. 1(b). To clarify the effect of the heat loss caused by heat transfer through the thermocouple on the measurement of *μ* and *S*, we consider the thermal conductance of the thermocouple creating a heat sink at the mid-point of the sample, which will reduce the magnitude of the temperature change induced by the Thomson effect. It is practically not easy to place the thermocouple at the exact mid-point of the sample. We thus consider the effect of the small displacement of the thermocouple from the middle of the sample to clarify the acceptable displacement.

When a dc current is applied, the temperature distribution *T*(*x*) can be determined by solving the following heat transfer equations:

where *ρ, κ*, *a*, *l*_{1}, *l*_{2}, and *I* on the left-hand side of (3) correspond to the electrical resistivity, thermal conductivity, cross-sectional area, sample length from the middle of the sample that satisfies the condition *l*_{1} + *l*_{2} = 2*l*, and dc-current amplitude, respectively. The first term on the left-hand side of (3) represents the heat conduction in the sample, while the second term denotes the heat generation caused by the Thomson effect, which is the heat absorption or generation occurring in addition to the Joule effect when an electric current flows along a conductor with a temperature gradient. Unlike Joule heating, the degree of Thomson heating or absorption depends on the polarity of the currents. The third term on the left-hand side of (3) represents Joule heating.

The temperatures at each end of the sample (*T*_{L} and *T*_{H}) are maintained constant, as expressed in (4). The thermal conductance of the thermocouple creates a heat sink at the mid-point of the sample, again as expressed in (4). Based on these constraints, (3) is solved over the intervals –*l*_{1} < *x* < 0, and 0 < *x* < *l*_{2} subject to the boundary conditions expressed in (4), where *K* denotes the thermal conductance of the thermocouple. Upon setting the conditions *l*_{1} + *l*_{2} = 2*l*, and *l*_{1} - *l*_{2} = 2*δl*, the temperature of the sample when the thermocouple is placed in the vicinity of the middle of the sample may be expressed as

Here, *C* and *D* represent the coefficients related to the Thomson and Joule effects, which are defined as *C* ≡ -*μΙ*/*ακ* and *D* ≡ *I*^{2}*ρ*/*a*^{2}*κ*, respectively. Further, *N*, the ratio of heat conductance in the sample to that in the thermocouple, is defined as *N* ≡ (1 + *K*/*K*_{0}), with *K*_{0} = 2*a*κ*/l*. The first, second, and third terms on the right hand side of (5) represent the temperature increase owing to the Joule effect, second-order Thomson effect, and first-order Thomson effect, respectively. We next analyze the application of ac current under the same boundary conditions. If the frequency of the ac current is sufficiently large, the Thomson heating effect can be neglected. Upon setting *C* = 0 in (5) and neglecting higher-order terms of the displacement from the middle of the sample *δl*/*l*, the ac-current-induced temperature increase can be simply expressed as

Joule heating and second-order Thomson heating can be canceled out by reversing the polarity of the dc current, since both effects are independent of the current direction. Upon calculating the average value of the temperature using (5) when a positive and negative dc current is applied, the relation between the Thomson coefficient *μ* considering the heat loss from the sample and the small displacement δ*l* of the thermocouple can be derived as

where Δ*T*_{dc} due to the Thomson effect is calculated as Δ*T*_{dc} ≡ (*T*_{+dc} - *T*_{-dc})/2. Here, *T*_{+dc}, and *T*_{-dc} correspond to, respectively, the temperature of the sample when positive and negative dc currents are applied in (5). Further, Δ*T* denotes the temperature difference of the sample calculated as Δ*T* ≡ *T*_{H} – *T*_{L}. We note that the Thomson coefficient in (7) includes the thermal conductivity, geometry of the sample, and correction term *K*/*K*_{0} related to the heat loss from the sample via the thermocouple even if the thermocouple is placed at the middle of the sample, i.e., δ*l* << *l*. When the thermal conductance of the thermocouple is considerably smaller than that of the sample, that is, *K* << *K*_{0}, (7) can be further approximated as

which corresponds to the conventional formula of the dc technique.^{12} When the conventional formula (8) is utilized to determine *S*, the diameter, length, and thermal conductivity of the thermocouple should be carefully considered to satisfy the condition *K* << *K*_{0}, along with the displacement of the thermocouple δ*l*. Upon rearranging (7), substituting (6) into (7), and setting *R* = 2ρ*l*/*a*, where *R* denotes the dc electrical resistance of the sample, the Thomson coefficient can be modified as

When the displacement of the thermocouple from the mid-point of the sample, *δl*, is significantly smaller than that of the sample half-length *l*, (9) can be approximated as,

which corresponds to the formula of the ac-dc technique.^{13} Therefore, the sample thermal conductivity and geometry can be replaced by the precisely and readily measurable parameters of electrical resistance and temperature change caused by the Joule effect in the ac-dc technique. Furthermore, the correction term *K*/*K*_{0} in (7) also disappears regardless of the heat loss from the sample through the thermocouple upon calculating the ratio of the temperature changes Δ*T*_{dc} and Δ*T*_{ac}. Therefore, this analysis suggests that the ac-dc technique is insensitive to systematic errors incurred in the determination of the material thermal conductivity, geometry, and heat loss from the sample owing to heat transfer via the thermocouple. As the ac-dc technique is insensitive to the thermal conductivity and geometry of the sample the ac-dc technique has potential application not for metals but also for various thermoelectric materials such as bismuth telluride. However, the small displacement of the thermocouple from the mid-point of the sample can still affect the measurement results even in the ac-dc technique, as indicated by (9). This source of the error should be treated as the uncertainty of the measurement.

## III. EXPERIMENTS

Figure 1(c) shows the block diagram of the experimental configuration used to measure the Thomson coefficient in this study. The sample stage was mounted on a cryostat developed for the measurement of the temperature dependence of the Thomson coefficient, which enabled us to control the temperature in a very stable manner compared with that of our previous room-temperature apparatus.^{13} This cryostat made it possible to perform high-sensitive measurements of the Thomson heat with a smaller temperature difference in the temperature range from 80 K to 320 K. The measured sample was composed of Pt (99.999%) in the form of a thin wire with diameter 0.5 mm and length 25 mm. The sample was clamped between pairs of massive polyimide-membrane-insulated Cu blocks. These blocks were designed as large isothermal areas for anchoring the sample ends, and the block temperatures were measured with a resistance thermometer for obtaining the temperature difference. The temperature difference was maintained at a small value of 3 K. To measure the temperature change at the middle of the sample, a thermocouple made of copper and constantan wire was attached to the sample by using varnish or epoxy glue. The reference ends of the thermocouple were the large Cu blocks maintained at *T*_{L}, where *T*_{L} is the temperature of the cold side Cu block. The edges of the thermocouple were electrically insulated. The thermocouple wire diameter of 50 μm or less was chosen in order to reduce heat conduction. A nanovoltmeter (Keysight 34420A) with a resolution of 100 pV was used as a detector to measure the voltage from the thermocouple. The typical temperature increase caused by Joule heating was <1 K, and the typical temperature change caused by the Thomson effect was <10 mK. The reading of the nanovoltmeter for the thermocouple response caused by the Thomson effect was approximately 100 nV or less. The measured drift of the voltage from the thermocouple was calculated to be 20 pV/h, which is an acceptable level. To obtain the precise DC components of the voltage from the thermocouple, the integration period of the nanovoltmeter was selected to cover an integer number of signal periods of the thermocouple in the sample. The sample stage was evacuated to pressures below 10^{-5} Pa to enable stable temperature control. The sample was covered with a temperature-controlled thermal radiation shield to maintain a very small temperature difference of 1 K between the middle of the sample and the thermal radiation shield. The sample current was provided by a high-precision ac and dc semiconductor source (Fluke 5520A). Ac and dc currents with an rms amplitude of 0.5 A and a set frequency of 62.5 Hz were applied alternately. The measurement frequency was determined by the sweeping the frequency of the ac and dc semiconductor source from 10 Hz to 100 kHz. The measured Thomson coefficient data were observed to be flat in the frequency range 10 Hz to 1 kHz. The accuracy of the ac and dc currents was less than 40 μA/A (ppm). The dc electrical resistance *R* was determined from the dc current and voltage across the sample by means of a high-resolution digital voltmeter (Keysight 3458A).

## IV. RESULTS AND DISCUSSION

The measured temperature dependences of the Thomson coefficient and that normalized with respect to the absolute temperature over 90 to 300 K are shown in Fig. 2 and its inset, respectively. The sample geometry including the length and cross-sectional area was measured. The value of the thermal conductivity of a thin Pt wire used for the conventional method is 71.6 W/mK at 300 K.^{15} From these measured data, we obtained a *μ* value of -8.6 μV/K ± 2.2 μV/K at a temperature of 300 K using the conventional formula expressed as (8). The error bars denote the expanded uncertainty (2σ) for the Thomson coefficient measurement. The detailed analysis results of the measurement uncertainty are discussed below. Next, the Thomson coefficient was calculated with the use of our formula (10) from the obtained electrical resistance, temperature change during ac and dc excitations, and temperature difference between the sample ends. The temperature increase caused by Joule heating was less than 1 K and the typical temperature change caused by the Thomson effect was less than 10 mK over the whole temperature range. We obtained a *μ* value of -8.6 μV/K ± 0.4 μV/K at a temperature of 300 K. This value obtained using the ac-dc technique is closely consistent with that obtained by the conventional dc technique with their uncertainties, regardless of the thermal conductivity and sample geometry.

To investigate the effect of the heat loss from the sample via the thermocouple, we measured the Thomson coefficient as a function of the diameter of the thermocouple at room temperature, i.e., thermal conductance of the thermocouple. Figure 3 shows the effect of the thermocouple wire size on the measurement of the Thomson coefficient using the ac-dc technique as well as the conventional method. Both experimental data closely agree for diameters less than 50 μm, wherein the heat loss owing to the heat transfer via the thermocouple is acceptable. However, the absolute value of the measured *μ* obtained by the conventional method decreases beyond diameters of 125 μm. These data indicate that the acceptable diameter of the thermocouple to be used for the conventional dc technique can be determined from the ratio of heat conductance in the sample to that in the thermocouple, expressed as *K*/*K*_{0} in (7). Meanwhile, the dependence of the data on the diameter of the thermocouple exhibits an improvement between 125 μm and 500 μm upon usage of the ac-dc technique. This is because the ac-dc technique compensates for the linear heat loss owing to heat transfer via the thermocouple via calculation of the ratio of the measured temperature changes for both ac and dc excitations as shown in (10). From these data, it is likely that the ac-dc technique is insensitive to heat loss via the thermocouple, which makes it possible to use a thicker thermocouple that is even easier to be attached.

The sources of uncertainty affecting the value of the Thomson coefficient are listed in Table I. As dominant uncertainties, the sample geometry, thermal conductivity, and temperature difference along the sample in deriving the Thomson coefficient are considered. The relative standard uncertainty of the thermal conductivity measurement is likely to be 5%.^{16} The temperatures of both sides of the sample were measured by means of a high-precision resistance thermometer, and subsequently, the temperature difference and averaged sample temperature were obtained. The standard uncertainty caused by the displacement of the thermocouple from the middle of the sample was calculated using (7) and (9). The contribution to the overall uncertainty seems to be small compared with other sources such as the geometry, temperature difference, and thermal conductivity. The measured standard deviation of the mean (1σ) for 5 measurements was 0.1 μV/K. The standard deviation of the mean and the systematic uncertainty components mentioned above were subsequently combined using the root sum of squares (RSS) approach. The overall expanded uncertainty for the absolute Seebeck coefficient measurement was subsequently determined for a 95% confidence level (2σ). An analysis of these uncertainty sources led to an expanded relative uncertainty as large as 26% for dc technique. In contrast, since the Thomson coefficient measured by the ac-dc technique is insensitive to the sample geometry and thermal conductivity as in the conventional dc method, this analysis led to an expanded uncertainty of 6%.

. | dc technique . | ac-dc technique . | ||
---|---|---|---|---|

. | . | Relative standard . | . | Relative standard . |

Parameters . | Uncertainty . | uncertainty (%) . | Uncertainty . | uncertainty (%) . |

ac current | — | — | 0.02 mA | — |

Cross-sectional area | 50 μm | 12 | — | — |

dc current | 0.01 mA | — | 0.01 mA | — |

Electrical resistance | — | — | 0.01 mΩ | — |

Length | 50 μm | — | — | — |

Sensor accuracy including the | 50 mK | 1 | 50 mK | 1 |

analog circuit noise (hot side) | ||||

Sensor accuracy including the | 50 mK | 1 | 50 mK | 1 |

analog circuit noise (cold side) | ||||

Thermal conductivity | 6 W/mK | 5 | — | — |

Displacement of the thermocouple | 320 μm | 0.1 | 320 μm | 0.1 |

Repeated measurement | 0.1 μV/K | 2 | 0.1 μV/K | 2 |

Combined uncertainty (1σ) | — | 13 | — | 3 |

Expanded uncertainty (2σ) | — | 26 | — | 6 |

. | dc technique . | ac-dc technique . | ||
---|---|---|---|---|

. | . | Relative standard . | . | Relative standard . |

Parameters . | Uncertainty . | uncertainty (%) . | Uncertainty . | uncertainty (%) . |

ac current | — | — | 0.02 mA | — |

Cross-sectional area | 50 μm | 12 | — | — |

dc current | 0.01 mA | — | 0.01 mA | — |

Electrical resistance | — | — | 0.01 mΩ | — |

Length | 50 μm | — | — | — |

Sensor accuracy including the | 50 mK | 1 | 50 mK | 1 |

analog circuit noise (hot side) | ||||

Sensor accuracy including the | 50 mK | 1 | 50 mK | 1 |

analog circuit noise (cold side) | ||||

Thermal conductivity | 6 W/mK | 5 | — | — |

Displacement of the thermocouple | 320 μm | 0.1 | 320 μm | 0.1 |

Repeated measurement | 0.1 μV/K | 2 | 0.1 μV/K | 2 |

Combined uncertainty (1σ) | — | 13 | — | 3 |

Expanded uncertainty (2σ) | — | 26 | — | 6 |

Once the Thomson coefficient was obtained over the temperature range of the interest, we computed the absolute Seebeck coefficient by using (2). Computation of the *S* value from the measured Thomson coefficient data involved the following steps: The value obtained from the superconductor reference (YBCO) at 85 K was taken as the starting point, and subsequent calculations were performed by integrating the measured Thomson coefficient normalized with the temperature under the curve in Fig. 2. The measurement uncertainty of the absolute Seebeck coefficient using the superconductor reference was also considered. Figures 4(a) and (b) show the absolute Seebeck coefficient values obtained by the ac-dc technique and the calculated expanded uncertainty (2σ), respectively. Increased uncertainties at higher temperatures reflect the propagation of the uncertainty in the temperature-normalized Thomson coefficient *μ*/*T*. The expanded uncertainty obtained with the conventional dc technique was more than 0.7 μV/K with a relative expanded uncertainty of 17% at 300 K. In contrast, we obtained a value of -4.8 μV/K ± 0.2 μV/K with a relative expanded uncertainty of 4% using the ac-dc technique at approximately 300 K. Our value is closely consistent with the values obtained by the conventional method over the temperature range within the measurement uncertainty, but is about four times more precise. This result was also compared with the current recommended *S* value of Pt.^{3,8} As shown in the enlarged view of Fig. 4(a), the mean value obtained with our technique differs by 0.1 μV/K, corresponding to about 0.5 combined uncertainty deviations from the current recommended value, *S* = 4.9 μV/K at 300 K.

## V. CONCLUSIONS

We proposed a measurement technique to precisely measure *S* using the Thomson effect. Our proposed ac-dc technique eliminates the need for acquiring accurate thermal conductivity and geometry values of the sample over a wide temperature range via the application of accurate equivalent-amplitude ac and dc currents in contrast to the conventional dc technique. The parameters of thermal conductivity and sample geometry are replaced by the precisely and readily measurable parameters of electrical resistance and temperature change caused by the Joule effect. The correction term of the heat loss owing to heat transfer via the thermocouple vanishes upon calculating the ratio of the measured temperature changes for both ac and dc excitations, unlike in the conventional method. This feature of our technique improves the precision of the conventional dc technique to an acceptable level, which is by a factor of four. Improving the precision and reliability of *S* measurements is important from the perspective of pure metrological interest and because of the key role that *S* plays in the study of thermoelectric materials, development of reliable thermoelectric standard materials, and precision high-temperature measurements.

## ACKNOWLEDGMENTS

This work was supported by JSPS KAKENHI Grant Number 15K21659. The authors express their gratitude to A. Yamamoto of ETRI/AIST and M. Maruyama of NMIJ/AIST, whose meticulous comments and stimulating discussions were of enormous help, and to A. Ichinose for proofreading the manuscript.