Introduction

Thermoelectric materials and elements (TEs) that can convert a temperature gradient into electricity (Seebeck effect) or electricity into a temperature gradient (Peltier effect) have drawn significant attention as a key technology for renewable energy1,2. The performance and energy efficiency of TEs at the temperature T have been described as a function of dimensionless figure of merit zT, where z (= S2/ρκ) is the function of Seebeck coefficient (S), resistivity (ρ), and thermal conductivity (κ)1,2,3. The zT values are typically estimated using two different TEs such as a rectangular solid for the measurement of S and ρ, and a thin disk for the measurement of κ3; however, the ideal estimation of the zT requires the use of the same TE or identical material. A direct zT measurement technique using a rectangular solid of the TE was proposed by Harman et al. in 1958. The Harman method4,5,6 uses AC resistance RAC with an alternating current (AC) and DC resistance RDC with a direct current (DC), based on which zT is expressed as zT = RDC/RAC − 1. However, the applicability of the method is limited owing to lack of information on the frequency of the AC and suitable magnitude of the AC and DC into the TEs.

Another approach to directly estimate zT using the same TE is a technique called impedance spectroscopy (IS), which uses the frequency domain based on an one-dimensional heat conduction equation7,8,9,10,11,12,13,14,15,16. Figure 1a,b show the schematic of Nyquist plot and the relation between its angular frequency (ω) and impedance Z(ω), based on which the zT is expressed as10,16

$$zT = \frac{{\left. {Z\left( {\omega \to 0} \right)} \right|_{{Q_{P} > > Q_{J} }} }}{{\left. {Z\left( {\omega \to \infty } \right)} \right|_{{Q_{P} > > Q_{J} }} }} - 1 = \frac{{R_{ohm} + \left. {R_{TE} \left( {1 + \frac{{\eta Q_{J} }}{{Q_{P} }}} \right)} \right|_{{Q_{P} > > Q_{J} }} }}{{R_{ohm} + \left. {\frac{{R_{TE} }}{4}\frac{{Q_{J} }}{{Q_{P} }}} \right|_{{Q_{P} > > Q_{J} }} }} - 1 = \frac{{R_{ohm} + R_{TE} }}{{R_{ohm} }} - 1 = \frac{{R_{TE} }}{{R_{ohm} }} = \frac{{\frac{{S^{2} T}}{\kappa }\frac{L}{A}}}{{\rho \frac{L}{A}}} = \frac{{S^{2} }}{\rho \kappa }T,$$
(1)

where Rohm and RTE are ohmic and thermoelectric resistance, QP ( =|S|TI) and QJ (= RohmI2) are Peltier and Joule heat, A and L are cross- sectional area and length of the TE, respectively, and η is a proportional factor associated with the heat flow of QJ to one side of the TE. In addition, zT (= RTE/Rohm) is denoted by the ratio of certain physical parameters such as resistance because zT is dimensionless12,16. Therefore, the physical meaning of the zT implies a ratio of the ohmic resistance (Rohm) to increasing resistance (RTE) generated by the temperature difference (ΔT) between the edges of the TE, which, in turn, is caused by the Peltier heat induced by DC. In particular, measuring the resistance, which is a macro physical quantity, is easy using recent electrometric instruments with a combination of a voltmeter for DC measurement and a lock-in amplifier for AC measurement by a precision current source. Equation (1) also shows that the condition QP >  > QJ must be met to obtain the precise value of zT. Therefore, the optimum current Iopt should be Iopt < <|S|T/Rohm. Moreover, the IS method is a suitable technique to determine the zT for both TEs and thermoelectric modules (TMs), which are an assembly of thermoelectric elements10,11,16. However, measuring the Z(ω → 0) takes several hours because a suitable characteristic frequency (ωTE) is required at ω → 0 (or ω <  < ωTE). ωTE is denoted as a function of thermal diffusivity (α) and L of the TE, ωTE ∝ α/L2, and typically approaches 1 rad/s owing to the small value of κ (∝ α) of the TEs. Therefore, the angular frequency satisfying Z(ω → 0) would approximately be of the order of 10–2–10–4 rad/s, depending on L12,15. The theory and model of the IS method clearly demonstrates that the Harman method is one of the results obtained using Z(ω → ∞) → RAC and Z(ω → 0) → RDC from Eq. (1) at QP >  > QJ16. Furthermore, the R2C approximation was applied to roughly explain the angular frequency dependence ZR2C(ω), as shown in Fig. 1a,b, using optimum current Iopt, which is expressed as12,16

$$Z_{R2C} \left( \omega \right) = R_{ohm} + \left. {R_{TE} \left( {1 + \frac{{\eta Q_{J} }}{{Q_{P} }}} \right)} \right|_{{Q_{P} > > Q_{J} }} \left( {\frac{{1 - \left( {{{j\omega } \mathord{\left/ {\vphantom {{j\omega } {\omega_{R2C} }}} \right. \kern-\nulldelimiterspace} {\omega_{R2C} }}} \right)}}{{1 + \left( {{\omega \mathord{\left/ {\vphantom {\omega {\omega_{R2C} }}} \right. \kern-\nulldelimiterspace} {\omega_{R2C} }}} \right)^{2} }}} \right)= R_{ohm} + R_{TE} \left( {\frac{{1 - \left( {{{j\omega } \mathord{\left/ {\vphantom {{j\omega } {\omega_{R2C} }}} \right. \kern-\nulldelimiterspace} {\omega_{R2C} }}} \right)}}{{1 + \left( {{\omega \mathord{\left/ {\vphantom {\omega {\omega_{R2C} }}} \right. \kern-\nulldelimiterspace} {\omega_{R2C} }}} \right)^{2} }}} \right)$$
(2)
Figure 1
figure 1

Schematics of (a) a Nyquist plot and (b) a frequency dependence using the IS method with Z(ω) and ZR2C(ω)10,12,16, and (c) time dependence of measured resistance R(t) using the TH method. An inset of (c) shows a schematic for the setting view of the TM prepared. (d) An equivalent circuit for thermoelectric element and thermoelectric module16.

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The transient Harman (TH) method, which is an alternative technique derived from the Harman method, is based on the transient response of resistance R(t) of the TEs and TMs using time domain, as shown in Fig. 1c. The TH method is relatively simpler to apply in the determination of zT using results of the R(t); several researchers have also reported its applicability17,18,19,20,21,22. In this method, Rohm and Rohm + RTE correspond to R(t = 0) and R(t → ∞), respectively, and zT is expressed as zT = R(t → ∞)/R(t = 0) − 1 using Eq. (1). The IS and TH methods (or R2C approximation) show that an equivalent circuit of the TE and TM can be expressed using three components, namely, Rohm, RTE, and CTE (called thermoelectric capacity), as shown in Fig. 1d. CTE is related to heat capacity of not only the TE(s) but also other components constituting the TE(s) like electrodes, especially for the TM10,12,16.

In this study, we comprehensively investigate several techniques of direct zT estimation based on the IS and TH methods. We employ a commercial base Π-shaped TM composed of bismuth-telluride (BiTe) (inset in Fig. 1c) to avoid influences of heat leakages through lead-wires attached for measurement and their contact resistance23. Furthermore, we highlight the disadvantages of both methods and suggest new techniques to overcome the drawbacks of the conventional methods. Finally, we propose a suitable technique to determine the value of zT precisely and directly within several minutes using a combination of AC and DC electrometric instruments, called time domain impedance spectroscopy (TDIS), and discuss the important factors required to obtain the value of zT precisely through measurements.

Results

Figure 2 shows the frequency dependence of the measured impedance Zmea(ω) in the case of the IS method for a TM prepared at 300 K with various AC from I = 100 μArms to 100 mArms16, and the inset shows its Nyquist plot using 1 mArms. The characteristic angular frequency (ωR2C) using the R2C approximation given by Eq. (2) was 0.255 rad/s (= 40.6 mHz) owing to L = 1.4 mm (ωR2C ∝ 1/L2). (Rohm)IS = Zmea(ω → ∞) = Re[Zmea(f = 513 Hz)] at ω/ωR2C ~ 104 (or phase angle |ϕ|< 0.1°) was 480.0 mΩ, and (Rohm + RTE)IS = Zmea(ω → 0) = Re[Zmea(f = 0.5 mHz)] at ω/ωR2C ~ 10–2 (or |ϕ|< 0.1°) was 869.5 mΩ at a current less than 10 mArms, satisfying QP >  > QJ. At 100 mArms, the contribution of QJ affected Re[Zmea(ω)] and − Im[Zmea(ω)] in the lower frequency region. Therefore, Re[Zmea(ω)] (∝ I) increased marginally at 10–3 Hz owing to QP (= 6.9 mW) ~ QJ (= 4.8 mW) using the representative magnitude of S (= − 231 μV/K at 300 K) corresponding to BiTe standard material24. In addition, ϕ = tan−1(Im[Zmea(ω)]/Re[Zmea(ω)]) helps identify the suitable frequency for determining Zmea(ω → ∞) and Zmea(ω → 0) because ωR2C is unknown15. Finally, using the IS method, the value of zT was clearly estimated as (zT)IS = 0.811 (= (Rohm + RTE)IS/(Rohm)IS − 1 = 869.5 / 480.0 − 1).

Figure 2
figure 2

Frequency dependence of impedance of real Re[Zmea(ω)] and imaginary part − Im[Zmea(ω)], respectively and phase angle ϕ at each AC (100 μArms to 100 mArms) for the TM prepared. The upper axis shows normalized angular frequency ω/ωR2C. A lock-in amplifier and Quasi-AC method (implemented using a high-precision AC source and digital multimeter using real-time data acquisition for the low-frequency region)12,28 were applied to measure the impedances at frequencies more than and less than 10 mHz, respectively. An inset shows its Nyquist plot of ZR2C(ω) and fitting plot by R2C approximation given in Eq. (2).

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Figure 3 shows the results of the TH method for the same TM used in the IS method at 300 K for transient current I(t). Rmea(t) was measured using a voltmeter (VM) and the remaining measurements were made using a data acquisition (DAQ) system. The inset in Fig. 3a shows the time dependence of Rmea(t) for different values of current ranging from − 500 mA to + 500 mA. In addition, the current dependencies of Rmea(t → ∞) at higher magnitudes of current due to the influence of QJ are clearly observable. I(t) using 1.447 mA at t ≥ 0 that is shown in Fig. 2a was selected from the essential condition of QP (= 100 μW) >  > QJ (= 1 μW) using (Rohm)IS. Figure 2b shows that the value of Rmea(t → ∞) = (Rohm + RTE)TH,VM = 869.3 mΩ measured by the voltmeter was specifically determined owing to the large signal-to-noise ratio (SNR). Rmea(t → ∞) was also measured at a different range of current, as shown in an inset in Fig. 3b, because RTE was replaced with RTE(1 + ηQJ/QP) in the higher current region in Eq. (1). It shows that Rmea(t → ∞) is proportional to I (∝ QJ/QP) in the higher current region, as expected16. To avoid aliasing of the signals measured by the voltmeter, the sampling rate was set as 2.3 Hz; consequently, Rmea(t → 0) = (Rohm)TH,VM was ambiguous. Therefore, data acquisition had to be performed at a higher sampling rate to detect the variation of Rmea(t → 0). The characteristic frequency ωR2C = 0.255 rad/s can be used as an approximate standard to derive the heat time constant τexp of the system (~ 4 s = 1/ωR2C). Figure 3c also shows Rmea(t) obtained using a DAQ system at a sampling rate of 100 kHz, which would be sufficient to detect the transient response of Rmea(t). Owing to the small SNR in the surroundings (or higher sampling rate), the data was averaged for a period ranging from 10 kHz to 1 Hz. However, detecting Rmea(t = 0) from the results of the TH method is difficult despite the use of the DAQ. As the expected value of Rmea(t = 0) was 480.0 mΩ from (Rohm)IS, the first data for Rmea(t = 0) was 471.27, 477.57, and 483.32 mΩ at 10 kHz, 1 kHz, and 100 Hz averaging, respectively. At a sampling rate of 100 kHz, the raw data of Rmea(t → 0) was distributed from 23.94 to 554.62 mΩ during 10 μs. This result shows that detecting Rohm using the raw data of the TH method is difficult despite using the higher DAQ system.

Figure 3
figure 3

Time dependence of transient response for (a) direct current I(t), measured resistance Rmea(t) and voltage Vmea(t) by (b) a voltmeter, and (c) a DAQ system at a sampling rate of using 100 kHz and its averaging results without error bar for each averaging period, respectively. The insets of (a) and (b) show the time dependence of Rmea(t) at each current and current dependence of Rmea(t → ∞), respectively.

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Another method was developed to determine Rohm and Rohm + RTE by fitting the formula into the data obtained by the DAQ system. This proposal was based on the fact that Rmea(t) is almost stabilized at 100 Hz averaging. Therefore, we used the value of Rmea(t) measured by the DAQ using 100 Hz averaging in the TH method. This approach was adopted based on our hypothesis that the average frequency enables us to represent the transient response with τexp. To estimate Rmea(t → 0) = (Rohm)TH,DAQ and Rmea(t → ∞) = (Rohm + RTE)TH,DAQ from Fig. 3c, two fitting equations for period Δt were applied from the equivalent circuit in Fig. 1d 16.

$$R_{R2C} (t) = \frac{{R_{ohm} + \left. {R_{TE} \left( {1 + \frac{{\eta Q_{J} }}{{Q_{P} }}} \right)} \right|_{{Q_{p} > > Q_{J} }} }}{{1 + \left. {\frac{{R_{TE} }}{{R_{ohm} }}\left( {1 + \frac{{\eta Q_{J} }}{{Q_{P} }}} \right)} \right|_{{Q_{p} > > Q_{J} }} \exp \left\{ { - \frac{t}{{\tau_{R2C} }}} \right\}}} = \frac{{R_{ohm} + R_{TE} }}{{1 + \frac{{R_{TE} }}{{R_{ohm} }}\exp \left\{ { - \frac{t}{{\tau_{R2C} }}} \right\}}},$$
(3)
$$R_{RC} (t) = R_{ohm} + \left. {R_{TE} \left( {1 + \frac{{\eta Q_{J} }}{{Q_{P} }}} \right)} \right|_{{Q_{p} > > Q_{J} }} \left( {1 - \exp \left\{ { - \frac{t}{{\tau_{RC} }}} \right\}} \right) = R_{ohm} + R_{TE} \left( {1 - \exp \left\{ { - \frac{t}{{\tau_{RC} }}} \right\}} \right),$$
(4)

where τR2C and τRC are the estimated time constants using each equation, respectively. Figure 4 shows the calculation results from the period Δt. Figure 4a,b show Rmea(t → 0) using the R2C (= (Rohm)TH,DAQ,R2C) and RC (= (Rohm)TH,DAQ,RC) approximations expressed in Eqs. (3) and (4), respectively. Near Δt ~ 0, both (Rohm)TH,DAQ,R2C in Fig. 4a and (Rohm)TH,DAQ,R2C in Fig. 4b were approximately 494 mΩ, and increased with increasing Δt owing to the excessive data available for Rmea(t). Finally, (Rohm)TH,DAQ,R2C and (Rohm)TH,DAQ,RC were asymptotically close to 524.6 mΩ in Fig. 4a and 504.5 mΩ in Fig. 4b, respectively, which are approximately 1.09 and 1.05 times higher compared with (Rohm)IS = 480.0 mΩ. The difference between (Rohm)TH,DAQ and (Rohm)IS obstructs the determination of (Rohm)TH,DAQ when estimating zT. In other words, the value of zT is underestimated because (Rohm)TH,DAQ > (Rohm)IS. Therefore, estimating (Rohm)TH,DAQ from Rmea(t → 0) obtained using the TH method is unsuitable even if the DAQ was used for measurement. Furthermore, Fig. 4c,d show that (Rohm + RTE)TH,DAQ converged at certain values, i.e., (Rohm + RTE)TH,DAQ,R2C = (Rohm + RTE)TH,DAQ,RC = 871.6 mΩ in Fig. 4c,d, which are consistent with (Rohm + RTE)IS = 869.5 mΩ. The difference is acceptable because the difference is approximately 2.1 mΩ, which corresponds to the approximate 3 μV difference during the DC measurement. Moreover, the estimated time constants τR2C and τRC were 3.27 and 4.06 s, respectively, neither of which matched the estimated value. However, in the RC approximation using Eq. (4), τRC = 4.06 s is not only close to τexp = 1/ωR2C = 4 s as a representative heat transport time in this system, but also simpler to apply. Figure 4f quantitatively shows that the required period using the normalized time Δt/τRC is more than 10 for each parameter. Furthermore, (zT)TH,DAQ was estimated to be 0.767 ± 0.014 (= (Rohm + RTE)TH,DAQ,RC/(Rohm)TH,DAQ,RC − 1 = 871.6/504.5 − 1), which is marginally smaller than that of the IS method because it is likely to overestimate (Rohm)TH,DAQ compared with (Rohm)IS.

Figure 4
figure 4

Estimated (a,b) (Rohm)TH,DAQ (= Rmea(t = 0)), (c,d) (Rohm + RTE)TH,DAQ (= Rmea(t → ∞)), and (e,f) time constant (τR2C and τRC) by R2C and RC approximation using the Eqs. (3) and (4) for the period Δt in Fig. 3c using the DAQ system (100 Hz averaging), respectively. The upper axes show the normalized time Δt/τR2C or Δt/τRC, respectively.

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We also attempted to precisely measure Rohm using the pulse DC with an identical DAQ system because the Rohm term adds the contribution of the temperature difference generated for the DC. Figure 5 shows the results of measuring Rmea(t → 0) as a function of t/Tp, where Tp is the pulse period. From Eqs. (1) and (4), the expression for Rmea(t → 0) at QP >  > QJ can be derived as

$$R_{mea} (t \to 0) \simeq R_{RC} (t \to 0) = R_{ohm} + R_{TE} \left( {1 - \exp \left\{ { - \frac{t}{{\tau_{RC} }}} \right\}} \right) \simeq R_{ohm} + R_{TE} \frac{t}{{\tau_{RC} }} = R_{ohm} \left( {1 + zT\frac{t}{{\tau_{RC} }}} \right).$$
(5)
Figure 5
figure 5

Normalized time dependence of (a) pulse current I(t/Tp), measured resistance Rmea(t/Tp) at (b) tp = 100 ms and Tp = 1000 ms, (c) tp = 10 ms and Tp = 100 ms, (d) tp = 1 ms and Tp = 10 ms for each pulse DC (1–10 mA ) by a DAQ system (100 kHz sampling rate), respectively. 10%, 40%, and 50% of the entire pulse period Tp in (a) correspond to the pulse DC width tp, relaxation period to remove the temperature gradient (or difference) on the TEs of the TM, and offset period to determine the zero voltage for the next measurement, respectively. The inset in (d) shows how (Rohm)pulse can be estimated from obtained data.

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Although the values of Rmea(t/Tp = 0) were approximately 480.0 mΩ (= (Rohm)IS) within the scattered data, it increased linearly at tp = 100 ms due to tp/τRC = 2.46 × 10–2, as expected from Eq. (5). Moreover, if the value of zT is large, typically 1, the term zT × tp/τRC in Eq. (5) should be less than 10–3 to ensure precise Rohm measurement. The increase in Rmea(t/Tp ~ 0) was suppressed at shorter tp, and the variation of Rmea(t) during tp was considered negligible owing to the considerably lower tp/τRC, as shown in Fig. 5d for tp/τRC = 2.46 × 10–4. Furthermore, current dependence was absent owing to a shorter tp, which possibly satisfies QP >  > QJ, during the tp/Tp cycle. Finally, although the use of pulse DC was expected to be suitable for measuring (Rohm)pulse, the measurement error within the scattered data near t/Tp ~ 0 was also considered.

Figure 6 shows the tp dependence of (Rohm)pulse, and fell within the accepted error margin. It quantitatively shows that (Rohm)pulse approached (Rohm)IS = 480.0 mΩ at tp/τRC < 10–3, as expected from Eq. (5). However, the measurement error of (Rohm)pulse was prominent at all tp owing to the low SNR. Moreover, (Rohm)pulse measured at a larger current, i.e., 10 mA, resulted in a smaller error compared with that measured at a smaller current; however, the error in the measurement of (Rohm)pulse was still large. Based on this evaluation, pulse and transient Harman (PTH) method was proposed for determining (zT)PTH using (Rohm)pulse and (Rohm + RTE)TH,DAQ. (zT)PTH was estimated to be 0.804 ± 0.043 (= (Rohm + RTE)TH,DAQ/(Rohm)pulse − 1 = 871.6/483.2 − 1) using (Rohm)pulse = 483.2 mΩ at I = 1 mA and tp = 5 ms (tp /τRC ~ 10–3), which is close to that of the IS method. Additionally, we attempted to measure (Rohm)delta using the delta method, in which a current source is synchronized with an oscillating square wave at a frequency (a kind of pulse current) using a voltmeter25. Subsequently, the resistance can be measured after eliminating offset voltage and removing the influences of the thermoelectric-motive force in the circuit. The details are summarized in Supplementary Information. Furthermore, an important conclusion was drawn from the measurements that were used to determine (Rohm)pulse using pulse DC in Figs. 5 and 6, (Rohm)delta in Fig. S1, and Rohm + RTE using DC in Fig. 3c by the PTH method. The precise measurement of Rohm from Rmea(t → 0) using only continuous DC is unsuitable for the TMs and TEs possessing large zT values despite the use of the DAQ system. This unsuitability can be attributed to the fact that large zT values affect the measurement noise (depending on the surroundings in the measurement system) even if the normalized pulse width tp/τRC is less than 10–3. Although the transported Peltier heat QP at I = 1 mA, shown Figs. 5 and 6 appeared small near QP = 69.3 μW ( =|S|TI = 231 × 10–6 × 300 × 1 × 10–3) at 300 K24, the Peltier heat flux qP across the TE was not negligible, i.e., qP = 164 W/m2 (= QP/A = 69.3 × 10–6 × (0.65 × 10–3)−2) for A = 0.65 × 0.65 mm2 owing to the generation of ΔT (~ qpL/κ).

Figure 6
figure 6

Pulse width (tp) dependence of estimated ohmic resistance (Rohm)pulse using 50-cycle averaging at each current. The upper axis shows normalized pulse width tp/τRC. The results using the R2C and RC approximation models with (Rohm)IS are also shown.

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Discussion

Based on the above-mentioned results and considerations, we proposed a combined measurement technique, namely, time domain impedance spectroscopy (TDIS), which attempts to measure Rohm = Zmea(ω → ∞) using a lock-in amplifier and AC at ω/ωR2C > 104 (or |ϕ|< 0.1°) and Rohm + RTE = Rmea(t → ∞) with a voltmeter. Alternatively, it employs a DAQ system using DC at t/τRC > 10 based on the TH method and partially derives the optimum current from the knowledge of the IS theory and model10,11,16. Consequently, (zT)TDIS was expressed as (zT)TDIS,VM = 0.811 ± 2.4 × 10–4 (= (Rohm + RTE)TH,VM/(Rohm)IS − 1 = 869.3/480 − 1) using the voltmeter or (zT)TDIS,DAQ = 0.816 ± 3.6 × 10–4 (= (Rohm + RTE)TH,DAQ,RC/(Rohm)IS − 1 = 871.6/480 − 1) using the DAQ (data of 100 Hz averaging). A marginal difference between (zT)TDIS,VM and (zT)TDIS,DAQ was derived from the measurement error of (Rohm + RTE)TH,VM and (Rohm + RTE)TH,DAQ,RC depending on SNR. The magnitude of the current Imax(t > 0) required to suitably measure Zmea(ω > 104 ωR2C) and Rmea(t > 10τRC) should fulfil the condition Imax <  < Rohm/(n|S|T) to satisfy QP >  > QJ, where n is number of TEs in the TM (n = 1 is for the TE). In short, although its concept is relatively similar to that of the original Harman method4, the TDIS method accidentally revealed the method to measure Zmea(ω → ∞) and Rmea(t → ∞) both quantitatively and qualitatively. Additionally, the temperature fluctuation ΔTf of the sample stage anchoring the TM during the measurements is an important factor. This ΔTf influences the measurement of the additional RTE (= n|SΔT|/I) from Rohm because the TDIS method is based on detecting ΔT derived from the Peltier effect (ΔT = RTEI/n|S|), given that ΔT >  > ΔTf. Furthermore, all the experiments in this study were performed at a standard deviation of ΔTf ~ 0.3 mK in high vacuum (~ 10–4 Pa) to maintain an adiabatic condition26,27. We hypothesize that ΔTf/ΔT (~ 0.3 mK/174 mK) is a key factor that determines how accurately the value of zT can be detected by the TDIS24.

Figure 7 and Table 1 show a summary of zT estimation using each technique. The (zT)IS estimated using the IS method, as shown in Fig. 2, was 0.811 ± 3.4 × 10–5. The result is reliable and (Rohm)IS (= Re[Zmea(f = 513 Hz)]) was obtained by the lock-in amplifier within several seconds; however, the measurement of (Rohm + RTE)IS (= Re[Zmea(f = 0.5 mHz)]) required several hours when employing the quasi-AC method in the lower frequency region (< 10 mHz) using the DC source and the voltmeter12,28. (zT)TH,DAQ estimated using the TH method, as shown in Figs. 3 and 4, was 0.767 ± 0.014. The estimated (zT)TH,DAQ differed from (zT)IS because determining (Rohm)TH,DAQ (= Rmea(t → 0)) was difficult despite applying the DAQ system. Finally, (zT)PTH estimated using the PTH method with pulse DC, as shown in Figs. 4, 5, and 6, was 0.804 ± 0.043, which was in moderately good agreement with (zT)IS. However, the measurement error to determine (Rohm)pulse was prominent. In the TDIS method, to quickly determine Rohm and Rohm + RTE without error, (Rohm)IS and (Rohm + RTE)TH,VM or (Rohm + RTE)TH,DAQ obtained from the lock-in amplifier and the voltmeter or the DAQ using the TH method with suitable AC and DC, respectively, were used. The zT values estimated by the voltmeter and the DAQ were (zT)TDIS,VM = 0.811 ± 2.4 × 10–4 and (zT)TDIS,DAQ = 0.816 ± 3.6 × 10–4, respectively. In addition, the error of (zT)TDIS,DAQ increased using raw data at a sampling rate of 100 kHz owing to the small SNR in Fig. 3d. We concluded that zT = ((Rohm + RTE)/Rohm − 1) can be determined by the TDIS method using Rohm = (Rohm)IS = Zmea(ω → ∞) measured by the lock-in amplifier and Rohm + RTE = (Rohm + RTE)TH,VM = Rmea(t → ∞) measured by the voltmeter within several minutes. Precise measurements were obtained directly using a combination of AC and DC electrometric measurements without any heat measurements.

Figure 7
figure 7

A summary of zT of the prepared TM estimated by various techniques, devices used, and conditions for Rohm and Rohm + RTE at 300.000 K.

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Table 1 A summary of techniques for direct zT estimation and its notes.
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In this study, we reported the zT estimation using the TM owing to its higher resistance (~ 1 Ω) compared with a TE (~ 1 to 10 mΩ). In the future, the measurement technique of the TDIS method will be applied to determine zT using any TEs with a definite geometry, such as a rectangular solid TE. Furthermore, we plan to establish the accuracy of the TDIS method using the temperature dependence of the zT of BiTe, which decreases with temperature in the lower temperature region12.

Conclusions

In this study, we developed a new method to directly estimate zT. The proposed method is based on the theory and model of the IS method using frequency domain with a Π-shaped TM for a BiTe system. However, the IS method possesses several drawbacks, such as the long time required to determine zT. The reason behind this drawback is that the information about both Zmea(ω → ∞) and Zmea(ω → 0) using frequency domain with AC is required. In addition, we used the TH method using the time domain with DC, which is based on the time dependence of the resistance Rmea(t), to measure the resistance Rmea(t → ∞) and Rmea(t = 0). However, the results showed that determining Rmea(t → 0) using the TH method is difficult. Furthermore, we attempted to estimate Rmea(t → 0) using the PTH method with pulse DC. However, we found that continuous DC was unsuitable for determining Rmea(t → 0) because determining the resistance derived from the electronic and Peltier heat at Rmea(t → 0) is difficult. To overcome these drawbacks, we proposed the TDIS method by combining the frequency and time domains to determine Rohm = Zmea(ω → ∞) using the lock-in amplifier with AC and Rohm + RTE = Rmea(t → ∞) using the voltmeter with DC. Finally, zT was estimated as zT = Rmea(t → ∞)/Zmea(ω → ∞) − 1 using optimum current Iopt that satisfies the condition QP (Peltier heat) >  > QJ (Joule heat), given that Iopt < <|S|T/Rohm. Furthermore, the estimated zT values of the TM using the IS and the TDIS methods were in perfect agreement, i.e., 0.811 at 300 K. Moreover, the TDIS method helped in qualitatively and quantitatively describing zT obtained from the IS method. We expect that this study will aid in developing more effective methods to determine zT precisely within several minutes for not only TMs but also any given TE.

Methods

A commercial-base Π-shaped thermoelectric module composed of BiTe was prepared (KSML007F, KELK). The total number (n) of the TEs for n- and p-types was n = 14. The impedance Zmea(ω) and resistance Rmea(t) were measured by four-probe method after attaching lead-wires to apply current and measure the voltage16. One side of the module was tightly fixed by a spring plate to a sample stage capable of controlling the temperature using a precise temperature control system (336, Lakeshore) at 300.000 ± 0.3 mK by calibrated Cernox thermo-sensor (Lakeshore) and two PID feedback heaters chilled by a cryo cooler (RDK-101D, Sumitomo Heavy Industry) under 10–4 Pa by vacuum pumps24,25. The frequency dependence of Zmea(ω) was measured by a lock-in amplifier (SR830, Stanford Research Systems) using an AC source (6221, Keithley) for frequencies higher than 10 mHz. Conversely, the quasi-AC method was employed using a DC source and a voltmeter (2182A, Keithley) for frequencies less than 10 mHz, implemented using a high-precision AC source and digital multimeter using real-time data acquisition for the low-frequency region12,28. The time dependence of the resistance Rmea(t) was measured using a DC and pulse current source (6221, Keithley) for currents less than 100 mA, and a DC source (2400, Keithley), voltmeter (2182A, Keithley), and DAQ system (USB-6281, NI) for currents higher than 100 mA. All the instruments were connected through GPIB and USB cables and controlled appropriately by the LabVIEW (NI) program.