Thermoelectric measurements are notoriously challenging. In this work, we outline new thermoelectric characterization methods that are experimentally more straightforward and provide much higher accuracy, reducing error by at least a factor of 2. Specifically, three novel measurement methodologies for thermal conductivity are detailed: steady‐state isothermal measurements, scanning hot probe, and lock‐in transient Harman technique. These three new measurement methodologies are validated using experimental measurement results from standards, as well as candidate materials for thermoelectric power generation. We review thermal conductivity measurement results from new half‐Heusler (ZrNiSn‐based) materials, as well as commercial (Bi,Sb)2(Te,Se)3 and mature PbTe samples. For devices, we show characterization of commercial (Bi,Sb)2(Te,Se)3 modules, precommercial PbTe/TAGS modules, and new high accuracy numerical device simulation of Skutterudite devices. Measurements are validated by comparison to well‐established standard reference materials, as well as evaluation of device performance, and comparison to theoretical prediction obtained using measurements of individual properties. The new measurement methodologies presented here provide a new, compelling, simple, and more accurate means of material characterization, providing better agreement with theory.
- thermal conductivity
- Seebeck coefficient
- electrical resistivity
- device efficiency
The efficiency with which a thermoelectric (TE) power generator can convert heat energy to electricity is determined, in part, by thermal conductivity,
In this work, we describe several new, more accurate techniques to measure thermal conductivity,
The truest test of the accuracy of measurement is comparison with fabricated devices. To support the validation of measurements of individual material properties, we outline a new device metrics, which allows comparison between theoretical and measured device efficiency. We outline a new
2. Novel measurements of thermal conductivity
2.1. Steady‐state isothermal technique
This new measurement of
In these past studies, principal parasitic heat flows causing error include conduction along lead wires, conduction along thermocouples, Joule heating within lead wires, and radiation. The magnitudes of these parasitic heat flows can be as large as 30% of Peltier heat. Penn quantified the significance of parasitic heat flows in Harman's technique, and showed, that they induced error of more than 10% . Bowley and Goldsmid , as well as Buist  reported, that parasitic heat flows cause error, usually larger than 20%.
The focus of the present work is description of a new, correctionless method to measure
Assume a sample having the temperature of one end anchored to the temperature of the environment,
To quantify the magnitude of
For progressively larger
From the requirement imposed by Eq. (3), a new method for measuring
2.1.1. Thermal conductivity of n‐type half‐Heusler
Thermal conductivity of a half‐Heusler alloy (Figure 3) was collected (triangles) and is presented with respect to previously published data (squares, circles, and diamonds) measured by the laser‐flash thermal diffusivity technique and reported by researchers from GMZ corporation . Because of the speed, ease, and simplicity of the new technique presented, there is opportunity for significantly more collected data. One data point can be collected in seconds. However, as can be seen in Figure 3, it is consistent to within experimental error and falls within the bounds of the published laser‐flash data.
2.1.2. Thermal conductivity of PbTe
One category of high performance thermoelectric materials is near‐degenerate semiconductors. Such materials do not directly obey the Wiedemann‐Franz relationship between electrical resistivity and thermal conductivity, due to the significant contribution of lattice thermal conduction to the total thermal conductivity. However, utilizing a modified Wiedemann‐Franz relationship to find the thermal conductivity due to electron flow allows direct, real‐time deconvolution of lattice thermal conductivity (
If thermal conductivity and electrical resistivity are measured, then
Figure 4 shows temperature dependences of PbTe thermal conductivity (including measured data) and deconvolution of electronic and lattice contributions to total thermal conductivity.
2.2. Scanning hot probe
The scanning hot probe technique provides measurement of local thermal conductivity and Seebeck coefficient of a sample by measuring average probe temperature when probe tip and sample are in “thermal contact”, i.e., when probe tip is in physical contact with the sample or is at known distance near enough to the sample to induce measurable heat exchange between probe and sample. Average probe temperature is also measured far from the sample, to account for the amount of heat lost to the surroundings and through the probe contacts. Difference in average probe temperature between these two cases is due to the heat transferred to the sample, which may be quantified through the following analytical derivation. Figure 5 depicts the thermal exchange between probe, sample, and surroundings, as well as the series thermal resistance network between probe and sample.
For steady‐state probe heating using DC current, (or AC current at low frequency, when the heat capacity effects are negligible, and temperature rise amplitude is frequency independent and equivalent to DC temperature rise to good approximation) the governing equation describing amplitude of the temperature profile of the probe shown in Figure 5 is given by :
where , (here, is the convective heat transfer coefficient, is the probe's emissivity, is an approximation for the exact (
To obtain an analytical solution to the second order differential equation, two boundary conditions are employed. The first assumption is that the ends of the probe are at ambient temperature (i.e., ). The second assumption is that the tip region of the probe of length 2
where the left‐hand side is heat conduction and Joule heating of the probe, is the probe's cross‐sectional area, and is the length of the probe, and where the right‐hand side is heat transfer through one leg of the probe (thus half the total heat transfer to the sample, by symmetry). Finally, heat transfer rate between probe and sample, , is:
where and are temperature of the probe and sample, respectively, at the tip region, and is samples thermal resistance. Solving Eq. (6) to obtain temperature profile along the probe for a given value of yields the following expression:
where , , and constants and are easily obtained by applying boundary condition .
If the sample is bulk, or has bulk‐like thickness, thermal conductivity is found from by employing semiinfinite medium assumption and 2D bulk sample assumption :
If the sample is a thin film of thickness on substrate, and is thin enough, that there is negligible heat spreading in the in‐plane directions of the sample, then thermal conductivity is found by solving the expression for the series thermal resistance across substrate and film, with 1D heat transfer across the thickness of the film :
When heat transfer may be multidimensional and anisotropic, models developed by Son et al.  for laser heating may be used to predict thermal resistance of the sample, based on the respective values of thermal conductivity for the film and substrate.
Data collected from scanning hot thermoelectric probe experiment are probe voltage, voltage across a reference resistor, Seebeck voltage, and photodetector voltage for position sensing. The value of current passing through the system is obtained by dividing voltage across the reference resistor by known electrical resistance of that resistor. Probe resistance is then found by dividing probe voltage by that value of current. Often, instead of single reference resistor, Wheatstone bridge is utilized. Figure 6 depicts the difference in circuit between measurement taken (a) with and (b) without Wheatstone bridge.
In DC mode, with resistor wired in series with the probe, measured probe voltage, , may be expressed in terms of voltage across reference resistor () as:
where is DC current passing through the circuit, is electrical resistance of the probe, and is known electrical resistance of reference resistor. Probe's electrical resistance is proportional to temperature rise above ambient, when the probe undergoes Joule heating. It may be expressed as:
where is average probe temperature rise, is nominal probe electrical resistance at 19.9°C, (not including electrical contacts to the circuit), when the probe is not being heated, is electrical resistance arise from contacts and circuit's wiring, and is probe's temperature coefficient of resistance, in terms of 1/°C.
Defining average probe thermal resistance as average probe temperature rise divided by Joule heating power, we can write average probe thermal resistance as:
Eq. (14) allows determination of by the slope of probe temperature rise with power applied, reducing the overall experimental uncertainty compared with a single value of temperature at a given power. If the circuit uses Wheatstone bridge, equations differ only by the method of finding electrical resistance of the probe. In this case, reduces to:
where and are voltages across bridge side and probe side, respectively. is the total resistance of bridge side of the circuit, and is no heating resistance of the probe side of the bridge. With the probe resistance obtained, remaining equations are left unchanged. When AC current of amplitude is passed through the circuit, then measured probe resistance can be expressed as:
The probe tip voltage is expressed as:
Thus, temperature amplitude is determined to be:
To obtain sample thermal conductivity, the probe must be calibrated. Quantities in Eqs. (6)–(11), which are not determined directly from experimental measurement are: , , , , , , and . To be fully calibrated, these quantities must be known. The probe manufacturer specifies values for , and , and these values are used in this work. Values and may be found by determining probe's geometry (typically from SEM or microscope images, but may also be determined by measuring in a vacuum and in air). is determined by measuring far from contact, and matching the value predicted by the analytical model by adjusting and integrating Eq. (9) from
|SiGe film on glass substrate||1.8 µm||2.8 ± 0.3 µm||14.948 ± 54 K/W||44.927±7820 K/W||76.134 ± 9494 K/W||1.22 ± 0.21 W/K·m // 1.23 ± 0.12 W/K·m|
|Fe‐doped PCDTBT (1:1 doping concentration)||3.0 µm||2.8 ± 0.3 µm||15.220 ± 155 K/W||44.927±7820 K/W||87.022 ± 14.631 K/W||1.03 ± 0.15 W/K·m |
|PCDTBT (non‐doped)||3.0 µm||2.8 ± 0.3 µm||17.866 ± 204 K/W||44.927±7820 K/W||358.859 ± 66.204 K/W||0.25 ± 0.04 W/K·m //0.20 ± 0.02 W/K·m |
|Tellurium Film||2.74 µm||2.8 ± 0.3 µm||15.749 ± 75.5 K/W||44.927±7820 K/W||112.476 ± 6.480 K/W||0.79 ± 0.04 W/K·m //0.78 ± 0.08 W/K·m |
|Au film on silicon substrate||150 nm||428 ± 24 nm||11.624 ± 157 K/W||40.191±1532 K/W||5505 ± 253 K/W||104.2 ± 67.4W/K·m //110 ± 2 W/ K·m |
|PEDOT CAL||Bulk||2.8 ± 0.3 µm||17.429 ± 217 K/W||44.927 ± 7820 K/W||241.732 ± 37.672 K/W||0.37 ± 0.05 W/K·m // 0.36 W/K·m|
|PANI‐5 % GNP CAL||Bulk||2.8 ± 0.3 µm||17.018 ± 115 K/W||44.927 ± 7820 K/W||188.595 ± 27.836 K/W||0.47 ± 0.06 W/K·m // 0.49 W/K·m |
|PANI‐7 % GNP CAL||Bulk||2.8 ± 0.3 µm||16.314 ± 118 K/W||44.927 ± 7820 K/W||131.760 ± 25.913 K/W||0.68 ± 0.08 W/K·m //0.65 W/ K·m |
|Bulk||2.8 ± 0.3 µm||15.700 ± 145 K/W||44.927 ± 7820 K/W||92.113 ± 11.911 K/W||0.97 ± 0.11 W/K·m // 1.0 W/K·m|
|Borosilicate Glass CAL||Bulk||2.8 ± 0.3 µm||15.516 ± 134 K/W||44.927 ± 7820 K/W||82.313 ± 9787 K/W||1.08 ± 0.11 W/K·m // 1.1 W/K·m|
|AISI 304 Steel CAL||Bulk||428 ± 24 nm||13.811 ± 119 K/W||40.191 ± 1532 K/W||37.511 ± 3511 K/W||15.6 ± 2.2 W/K·m // 16.2 W/K·m|
99.9 % pure Niobium CAL
|Bulk||428 ± 24 nm||12.194 ± 140 K/W||40.191 ± 1532 K/W||10.632 ± 2329 K/W||54.9 ± 8.9 W/K·m // 53.7 W/K·m|
2.3. Transient and lock‐in Harman techniques to decouple material ZT and thermoelectric properties
Finally, a new method of measuring material thermal conductivity by simultaneously measuring thermoelectric figure of merit (ZT), electrical resistivity (), and Seebeck coefficient () is proposed. ZT is dimensionless measure of the efficiency of material at converting thermal into electrical energy, or vice versa, at a given temperature,
Thus, thermal conductivity is obtained if the other terms in Eq. (19) are known. This new method also allows for measuring intrinsic ZT with reduced experimental error by accounting for losses through nonideal contacts and geometry.
Conventional application of Harman method uses four probes–two to pass current, and two to measure the voltage response of the sample. Harman demonstrated that, while electrical response of the sample was nearly instantaneous, voltage generated by Seebeck effect, which is thermally driven, is much slower. By taking advantage of this fact, thermal signal could be determined from voltage response of the sample to a sudden change in voltage over time, or response to an AC current passed, locking into thermally driven signal. It was shown, by letting , , and , that could be reduced to the ratio of resistive voltage to Seebeck voltage (
From resistive voltage, one may be able to determine electrical resistivity of the material; however, Seebeck coefficient and thermal conductivity remain coupled in equation for ZT. To decouple them, Seebeck coefficient may be simultaneously determined by adding a pair of thermocouple wires at the top and bottom surfaces of the sample as per Figure 9. If we label electric potential in each corner of the sample , respectively, using Ivory technique , we may find Seebeck coefficient from taking voltage measurements across the sample. If voltage is measured at opposite corners, then voltage values measured are and . Value is determined from expression below, where is the slope of vs :
This technique for measuring Seebeck coefficient reduces the required number of voltage measurements to determine from three to two and mitigates mismatch in thermocouples, since DC offsets are removed by using a slope. It also allows for AC measurements of the total voltage, and determination of ZT from Nyquist diagrams.
2.3.1. Transient Harman technique–analytical model
Several experimental setups used by the research community for thermoelectric characterization of thin films employ clean room microfabrication techniques to pattern a metallic electrode on top of the sample, while others use bonded wires or micromanipulated probes to make electrical contact with the top surface of the film sample [3, 21–24]. Configuration modeled in this work is similar to these situations, as shown in Figure 8. Thermoelectric film (3) with cross‐sectional area,
Eq. (21) represents steady state energy balance in layers 1–3; the first term on the left side represents heat conduction, the second represents lateral convection and the third represents Joule heating. Thermal radiation and temperature dependence of thermoelectric properties have been neglected as small temperature differences are assumed to occur during the experiments.
General solution of Eq. (21) for temperature profile in the layer
Integration constants c1, i and c2, i are determined using six boundary conditions. First, the temperature at free end of the probe of length
Second, the temperature of the end of the probe in contact with the surface is assumed to be constant temperature (see expression in the ensuing discussion, Eq. (31)).
The third boundary condition considers energy conservation at the interface between the film and the substrate:
In Eq. (24), the left side represents heat transfer rate out of the interface. It includes heat conduction and Peltier terms in layer 3, respectively. The middle section of Eq. (24) represents one way to express heat transfer rate entering the interface. It is written as the sum of heat conduction across the interface thermal contact resistance and contact Joule heating term deposited at the interface in layer 3. It is assumed, that the total contact Joule heating is split equally on both sides of the interface. The right side of Eq. (24) represents the second way to express heat transfer rate entering the interface and includes: (1) substrate heat conduction transfer rate written as the temperature difference across the substrate divided by substrate's thermal conduction resistance; (2) total Joule heating due to electrical contact resistance; and (3) Peltier contribution due to electric current flowing through the substrate electrode. It is assumed, that the bottom surface of the substrate is at ambient temperature. Thermal conduction resistance of the substrate, Θsubst, can be determined by conduction shape factor. For instance, for sample of diameter
Similar to Eq. (23), the fourth boundary condition is energy balance at the interface between probe and electrode:
Here, heat transfer rate exiting the interface (the left side of the equation) also includes convection from the top surface of the electrode to the ambient. Similar analysis is performed at the electrode‐sample interface, as stated in (26):
Finally, continuity of electrical current in the layers of the sample requires:
Modeling approach discussed above can be used to study in detail effects on temperature profile due to thermal and electrical properties of individual layers and contacts.
Thermal conductivity of the thermoelectric film is typically determined from relationship between temperature rise (usually the measured surface temperature) and dissipated power. In addition, difference between the surface and substrate temperature together with Seebeck voltage developed across the film is used to calculate Seebeck coefficient of the film. Practitioners in thermoelectric field need a way to evaluate steady state surface temperature before electrical current is switched off for transient Harman method under nonideal boundary conditions.
The main strategy pursued here is to use the superposition principle to calculate the total temperature rise by solving separately for temperature solutions under Joule heating and Peltier effects. Rather than using full set of Eqs. (21)–(27), several assumptions are made in this section in order to arrive at an easy to use expression for the surface temperature, which still reflects the main thermoelectric transport mechanisms in many practical situations. These assumptions are: (1) electrode's contributions (layer 2) to thermoelectric transport are neglected because metallic electrode layers typically have low Seebeck coefficient similar to the probe and much lower electrical and thermal resistances compared to thermoelectric films; (2) Seebeck coefficient of the current probe is neglected; (3) convection terms on the film surfaces are neglected; (4) substrate thermal resistance and film‐substrate electrode thermal contact resistances are neglected when compared to film thermal resistance, since thermoelectric film samples are typically low thermal conductivity films on high thermal conductivity substrates; and (5) Joule heating at film‐substrate electrode contact is neglected because under assumption (4) the substrate acts as a heat sink.
where the constant
Then, the solution for LTC of the top surface of the sample can be calculated as:
Heat transfer along the probe is calculated by solving the fin model with volumetric Joule heating and a temperature rise equal to zero (relative to the ambient) at free end of the fin (away from the sample). The equation for the probe heat transfer is:
where the constant
Then, the solution for NLTC is given by:
Finally, total temperature of the top surface of the sample, Ts, is:
where the first term contains Peltier effect's induced contributions to the surface temperature, the second term includes Joule heating effects from the sample and contact, and the third term includes Joule heating contribution from the probe wire.
Temperature of the probe at junction with the sample surface is then calculated as:
Understanding how to eliminate or reduce the effects due to heat loss and electrical and thermal contact resistances is critical in designing test structures amenable for accurate thermoelectric transport measurements. Parasitic effects are expected to be different for macroscale versus microscale samples, and this section focuses on microscale samples. To illustrate these effects, the surface temperature predictions as a function of current density are discussed for thermoelectric sample of 10×10×10 µm3 in contact with copper probe of 5 µm diameter and 1.3 mm length. Thermoelectric properties of thermoelectric film are similar to n‐type Bi2Te2.7Se0.3 and are listed in Table 2.
|Lateral dimension (μm)||76.2a||1014b||1014b|
|Seebeck coefficient (μV/K)||1.84||−212||1.68|
|Thermal conductivity (W/m K)||401||1.5||82|
|Convection heat transfer coefficient (W/m2K)||200c||10||10|
|Electrical resistivity (nOhm m)||17.1||10500||84|
Figure 10 shows rise of surface temperature with respect to ambient temperature, calculated from Eq. (42) for a range of specific thermal contact resistances. Electrical resistance of contact was assumed to be equal to theoretical limit predicted for electrical boundary resistance between Bi2Te3 and metal electrode . Direction of electrical current was chosen such, that the sample surface undergoes Peltier cooling. At low current densities, Peltier cooling term dominates over Joule heating terms and temperature of the top surface of the sample decreases linearly as electrical current density increases. After reaching the maximum cooling temperature at optimum current density, Joule heating terms start to dominate over Peltier terms. Parasitic conduction heat transfer effect is apparent even at very low current densities, as shown by inset in Figure 10. It leads to reduction of temperature difference across the sample as compared to predictions of an ideal Harman model. On the other hand, as thermal contact resistance increases, the sample cooling is stronger because the thermal barrier created at the contact reduces heat transfer rate with the probe. Importance of thermal barrier effect is gauged by comparison between thermal resistances of the probe, probe‐sample contact, and the sample itself. The modeled probe has thermal resistance of ∼5×104 K/W, which is similar to 6×104 K/W thermal resistance of the sample; therefore, a significant heat transfer occurs through the probe. As the thermal contact resistance increases, the probe heat transfer is reduced, particularly after contact thermal resistance becomes of the same order as thermal resistance of the probe. Alternative way to minimize the probe heat transfer rate could be realized by reducing diameter of the probe. However, besides practical challenges, this may have a negative impact associated with increase in resistance of electrical contact, as shown in Figure 10.
Heat transfer through the substrate could also play a major role in establishing the surface temperature. For large current densities, the strength of heat transfer through the substrate is indicated by large positive temperature difference measured across the sample. This difference is in contrast with the predictions of an ideal Harman model, where Joule heating effects never generate a temperature difference across the sample.
The effect of electrical contact resistance in absence of thermal contact resistance is investigated in Figure 10. The modeled probe has electrical resistance of ∼1 Ohm, which is similar to electrical resistance of the sample, therefore Joule heating effects can occur simultaneously in the sample and the probe. In addition, even for very low specific electrical contact resistance of 1×10‐11 Ohm×m2, electrical contact resistance for the modeled probe is considerable (0.5 Ohm). Electrical contact resistance increases by two orders of magnitude if the probe diameter is reduced by a factor of 10. This illustrates strong requirements to control the probe and contact electrical resistances in transient Harman experiments performed on film on substrate samples. This is because good thermoelectric samples have low electrical resistances, so Joule heating effect in contacts and probe can easily become dominant. Inspection of the probe and wire Joule heating terms suggests mitigation of the electrical contact resistance problem may be achieved by preparing thick film samples (large
One proposed strategy for determining the properties under nonideal conditions is the bipolar method, where transient Harman experiments are performed using direct and reversed current directions and where measured Seebeck and resistive voltages across the sample are averaged. This is believed to eliminate Joule heating effects and reveal the intrinsic Peltier effects in the sample [21, 23]. However, as demonstrated in this work, when nonideal boundary conditions are present, parasitic effects cannot be always completely eliminated by this strategy. Nevertheless, the analysis below demonstrates the ability to exploit this behavior to determine both thermal and electrical transport properties of the samples and their contacts.
Bipolar resistive voltage difference measured across the sample Δ
To find expression for bipolar Seebeck voltage, first Seebeck coefficient,
When bipolar method is used, then
Next, under small current approximations:
a simplified expression for bipolar surface temperature difference is obtained as:
Expression for ZT obtained through bipolar technique is found as:
Another strategy is to perform experiments over a range of currents and use differential changes in
Neither bipolar nor differential current methods alone are able to account for all parasitic effects. As a result, these effects must be considered in data reduction or otherwise minimized. A variable thickness method [21, 23, 24] is used to account for electrical contact resistance effects, while heat losses and thermal resistance effects are neglected. A different method to determine all thermoelectric properties without the need for extensive sample preparations is outlined below.
The strategy explored here is to use bipolar experiments performed over a wide range of currents rather, than small current regime required by above methods. It is expected, that at large currents, experimental Seebeck voltage and temperature signals become sensitive to electrical transport properties of the sample and contacts and could be used to determine the sample and contact thermoelectric properties. In addition to Seebeck and resistive voltage drops, method requires measurement of the sample surface temperature or the probe temperature (at the contact with the samples surface).
Proposed strategy takes into consideration selective sensitivity of thermal signals to Peltier and combined Peltier and Joule heating effects under low and high current regimes, respectively. Under small current approximations, temperatures of the probe and sample surface are linear with current, and thermal conductivity can be expressed as a function of experimentally measured slope of the probe temperature as:
In Eq. (52), value of Seebeck coefficient is substituted from Eq. (47), which is valid at any current. Next, Eqs. (44), (47), and (52) are substituted in Eq. (43). For the sake of discussion it is assumed, that specific electrical contact resistance is similar at the top and bottom contacts (other assumptions are discussed in Section 3). After the above substitutions, predicted probe temperature and Seebeck coefficient become a function of two unknowns, specific electrical and thermal contact resistances, which are then used to fit experimental signals under large current regime for both direct and reverse currents. This strategy allows the unique determination of all thermoelectric properties of the sample and electrical and thermal contact resistances. Details of the fitting procedure are presented in experimental validation section.
2.3.2. Lock‐in Harman technique–analytical model
The first term on the left side accounts for conduction through the wire/sample, where
Figure 11 represents the same configuration of sample and wires as considered before, but here shows heat transfer domains used in the model. 1D heat transfer was modeled in each of seven domains, one each for the six wires plus another for the sample. Details of the boundary conditions are given below. Temperature solution requires a total of 14 boundary conditions, two for the sample and four for each set of wires. The wire boundary conditions are as follows: two boundary conditions per wire (or six total) were defined by assuming, that the end of each wire was at room temperature, since the wires in the experiment were relatively long compared to their width and measurements were conducted in ambient conditions. The remaining two boundary conditions per wire (summing to twelve in total) are that the ends of each wire in contact with the sample are at a fixed temperature. The two boundary conditions across the sample come from the fact, that heat transfer at the interfaces must be balanced. For the interface between the first set of wires (domains 1–3) and the sample (domain 4), the energy balance yields Eq. (54):
The first term is the rate of heat conduction to and from the wires and is a function of temperature gradient at the wire‐sample interface and thermal conductivity of the wires. This is calculated using the pin fin equation below for the experimental results. The second term accounts for Joule heating due to electrical contact resistivity between the current lead and the sample,
Substituting this into governing equation and applying Fourier transform gives transformed governing equation:
Dirac delta function,
Joule heating term is again present at
The undetermined coefficients were solved numerically using transformed boundary conditions.
2.3.3. Experimental results–transient Harman
Figure 12 shows experimentally measured total and Seebeck voltages as a function of current.
Resistive voltage drop obtained after subtracting Seebeck voltage from total voltage includes contributions from the sample, probe‐sample contact, and sample‐substrate electrode contact. Inset in Figure 12 shows an example of measured voltage as a function of time during an experiment at 163 mA. The figure of merit calculated according to classical Harman method yields an average value of 0.11, much smaller than the manufacturer value of 0.85. This discrepancy is due to parasitic effects neglected in classical technique.
Measured probe temperature is linear with current for the smallest two bipolar currents and yields a slope of −55.1 K/A and was substituted in Eq. (52). Originally, only one probe was modeled in contact with the sample, the energy loss across constantan wire was evaluated and found to be in an order of magnitude smaller than for copper wire, therefore its effect is expected to be negligible.
Transient Harman experiments performed under the highest direct and reverse current conditions were used in the fitting of sample's thermoelectric properties and contact resistances. In the fitting procedure, thermal contact resistance
Since the original sample‐substrate interface from commercial Peltier device has a negligible electrical contact resistance compared with the sample,
The thermoelectric properties arise from each of the following four cases are summarized in Table 3. In case 1, electrical contact resistance
|Case #||Rth,1‐3 (K/W)||κ3 (W/mK)||(||Rc,1‐3 (m2)||(|
|1||1.1 × 10−6||1.38||−212||2.9 × 10−10||5.7×10‐5|
|2||6.6 × 10−7||1.48||−218||6.2 × 10−10||Nonphysical (negative)|
|3||9.4 × 10−7||1.43||−215||3.9 × 10−10||3.3 × 10−5|
|4||7.8 × 10−7||1.46||−217||4.9 × 10−10||1 × 10−5 (manufacturer specified)|
The highest deviations between measured sample's Seebeck and thermal conductivity as compared with manufacture's values are respectively 6 μV/K (3%) and 0.12 W/(m×K) (8 %). These deviations are smaller than the experimental uncertainty. The uncertainty in thermal conductivity due to propagation of the uncertainty in temperature and voltage measurements was calculated to be 0.26 W/(m×K). Similarly, for Seebeck coefficient, the uncertainty is equal to 9.9 μV/K. To accurately determine the sample resistivity, resistive voltage drop should be measured through the copper probe, which is used to pass electrical current, at the same time as through the constantan wire, so total electrical contact resistance and sample resistivity can be accurately determined. When correct resistivity of the sample was employed in the fitting, the sample thermal conductivity was within 3 % of manufacture's values.
Figure 13 shows comparison between measured and calculated temperature of the probe as a function of electrical current passed through it.
The theoretical predictions use the fitted thermoelectric properties and employ Eq. (43) with either all terms or only Peltier terms. The theoretical predictions were performed for all cases 1–4 and, since they superpose along the same line, they are not individually distinguishable in Figure 13. Joule heating effects are important at large currents in tested sample, as demonstrated by the discrepancy between Peltier heating only predictions and combined Peltier and Joule heating model. Predictions based on solving Eq. (20)–(27), that also include the convection on the sample surface and the contributions from indium electrode, show no significant difference with prediction based on Eq. (43). There is an excellent agreement between experimental and modeling data over entire electrical current range. Data sets for intermediate current values (∼85 mA) show also excellent agreement, although they have not been used in the fitting.
2.3.4. Experimental results–lock‐in Harman
ZT and individual thermoelectric properties may also be characterized experimentally using a Nyquist plot (plotting imaginary vs real parts of the complex voltage signal or sample temperature rise). Nyquist analysis of voltage measurements across the sample allows for direct calculation of the slope
The samples measured were bulk bismuth telluride alloys with dimensions of 4.5×3.8×3.8 mm3. Thin layer of gold was deposited on either end to improve adhesion and current spreading between the sample and lead wires. One lead wire and one thermocouple were soldered to either end of the sample. Current was applied through un‐insulated 50.8 µm diameter copper wire, and voltage was measured using 50.8 µm E‐type thermocouples. Two sets of voltage measurements were made across the sample using each set of thermocouple wires for excitation frequencies between 10 mHz and 10 Hz. Amplitudes of resulting voltages,
These two voltages were used to find Seebeck coefficient by Ivory's technique  using Eq. (21). Non‐imaginary values of two voltages are plotted against each other in Figure 15 and resulting in Seebeck coefficient αsample = 202.6 ± 1.4 μV/K. Real parts of signals are used, as these are components, that are in phase with excitation signal and as a result are in phase with each other. Amplitudes may be out of phase with each other and imaginary part is much smaller. Advantage of Ivory technique is that the magnitudes of measured voltages are greater than in traditional technique, if
Obtained data is again shown as superposition of resistive
To find equation for intrinsic ZT, the derivation of ZT performed by Harman can be repeated to include terms for heat loss and contact resistance. This adds two correction factors as shown in Eq. (62). The first is the ratio of heat lost from the sample to Peltier heat generation at the wire‐sample interface. This heat loss may be due to either convection or radiation from the end of the sample or conduction through the contacts. The second correction factor is the ratio of voltage drop across the contacts to that in the sample, which is equal to resistance of the contacts divided by that of the sample:
For this experiment, radiation and convection from the sample itself were negligible compared to heat loss through the contacts and only the latter was considered. Since ZT was calculated using voltages approximating DC, where heat transfer is in steady state, and high frequency AC, which is not affected by heat losses, steady‐state equation can be used to account for heat loss. Conduction through the wires is described by Eq. (63), which is the fin equation, where the base temperature is equal to that of the wire‐sample interface,
For long thin wires, the hyperbolic tangent goes to one and may be neglected. We considered losses by convection through the wire and temperature at the end of the wire away from the sample was assumed to have reached ambient (since wires were long and thin). Even though, the sample temperature was not measured directly, the temperature gradient across the sample was found from measured Seebeck coefficient value,
Samples resistivity was determined as 7.0×10−6 Ohm×m based on resistive voltage,
The same material properties were also found by fitting the predictions based on the numerical model described above in Eqs. (58)–(60) to experimental data. The fitting is shown in Figure 17, where Seebeck voltage data was converted to temperature amplitude using measured Seebeck coefficient. Adjusting thermal conductivity in the model and using the least squares fit, thermal conductivity was 1.55 W/(m×K) and thermal diffusivity was 9.5×10−3 cm2/s.
Overall, calculated uncertainty in this experiment was small with that in Seebeck coefficient and extrinsic ZT measurements being lower, than that of calculated intrinsic ZT. The latter is due to the uncertainty in calculating the heat loss through the wires, specifically calculating the heat transfer coefficient between the wires and the air. The heat transfer coefficient was calculated assuming a horizontal cylinder in air. While the uncertainty for the heat transfer coefficient was determined to be about 2%, its uncertainty was assumed to be closer to 10%. This was done because there is some additional uncertainty surrounding the assumptions of free convection and due to its dependence on lab conditions. The uncertainty could be improved by testing the sample in an evacuated chamber, eliminating entirely the need to calculate heat transfer coefficient. The uncertainty of extrinsic ZT is due to that of voltage measurements, and was assumed to be 1% of the measured value, and temperature, assumed to be 296 ± 2 K. The uncertainty in voltage measurement was assumed 1% as conservative estimate. The error of the device was lower, but noise in the system and variation between measurements was closer to 1%. Some error in Seebeck coefficient measurement will be present due to the assumption, that temperature gradients across all the wires in each thermocouple are identical. Since the junctions of thermocouples were somewhat embedded in solder, there may be slight temperature gradient between two wire‐solder interfaces. However, this difference was assumed to be negligible compared to temperature gradient along lengths of wires, and the uncertainty in Seebeck coefficient measurement was calculated as less than 1%. The uncertainty in
3. Verification strategies for measurements
3.1. Slope‐efficiency method: rapid measurement of device ZTmaximum.
Maximum electrical power output,
For TEG consisting of some number “i” of individual “thermocouples” connected in series and each having n‐type thermoelement and p‐type thermoelement, Seebeck effect relates
This expression highlights first important point:
The efficiency, Φ, with which TEG can convert heat flow,
Eq. (68) can be rewritten, assuming for simplicity a unicouple (i = 1), as:
The flow of heat is dominated by thermal conductivity of the materials from which TEG is constructed, so Fourier's law can be used to express
For planar TEG devices, the values of of both n‐type and p‐type thermoelements are equal; however, cross‐sectional areas of n‐type and p‐type may be quite different. Identifying cross‐sectional area of n‐type as
The proportionality between Φ and Δ
Note, that when area‐to‐length ratios are optimized for maximum efficiency, this relationship reduces to the common, well‐known expression for device ZT:
TEG efficiency can be measured as function of Δ
This expression highlights a second important point, that Φ should linearly increase as a function of Δ
A new index to determine maximum ZT of TEG device can be obtained by measuring the slope of TEG efficiency. To calculate maximum ZT, four times the slope of TEG efficiency multiplied by maximum temperature, under which TEG displays linear behavior with respect to Δ
The significance of this analysis is that it allows unique means to rapidly obtain ZTmaximum and confirm properties and individual measurements. Measurements can be confirmed by measuring slope of efficiency as function of Δ
3.1.1. Analysis of commercial (Bi,Sb)2(Te,Se)3 module
Efficiency of commercial (Bi,Sb)2(Te,Se)3 device is presented in Figure 18. This device is designed for high thermal impedance and has optimum performance window from nearly room‐temperature to roughly 425 K. Slope of efficiency was determined and is shown in inset of Figure 18, and is given as 0.0004/K. As expected, the slope is highly linear function of Δ
Obtained value of ZTmaximum is equal to 0.7, which is consistent with established values for commercial devices.
3.1.2. Analysis of PbTe/TAGS module
Efficiency of PbTe/TAGS device is presented below. Figure 19 shows temperature dependence of PbTe/TAGS module efficiency. At low temperature, the slope is somewhat a nonlinear function of Δ
3.2. Discretized heat‐balance model and analysis
More detailed device analysis and performance modeling including effects of temperature‐dependent material properties may be accomplished through the use of numerical methods. One technique for performing numerical analysis on TEG was reported by Lau and Buist  and later confirmed and expanded upon by Hogan and Shih . It involves partitioning the legs of TEG into virtual segments for computational purposes, where each segment is taken to be isothermal. Neighboring segments then vary in temperature such, that governing thermoelectric heat balance equations based on constant parameter theory are satisfied [28, 29]. This process is illustrated in Figure 20.
Based on constant‐parameter theory  and heat balance at the top surface of the ith segment, we have:
where the total heat flux
Here αi is Seebeck coefficient of the ith segment at temperature
where power P is delivered to the external load resistor,
The discrete heat balance equations (Eqs. (78) and (79)) derived in this manner can be easily solved for single leg of TEG with an iterative technique [26, 27]. For a given TC, an initial estimate is made for the heat delivered to the cold junction,
The initial hot‐side temperature of segment is taken as a uniform temperature for the entire segment and its thermoelectric properties are then determined from a curve fit to measured data. Adjacent segments attain different temperatures as the system is solved according to the energy balance requirements. Thus, temperature‐dependent effects are fully incorporated into the model. In fact, Hogan and Shih  were able to demonstrate excellent agreement using the discrete approximation as compared with an exact analysis of temperature‐dependent TEG performance by Sherman et.al. .
Figure 21 shows temperature profiles calculated for n‐type Skutterudite material with temperature‐dependent properties, operating at the indicated boundary conditions. For simplicity, electrical current is treated here as though there were an external load resistance matched to the internal resistance of the thermoelectric material leg, thus producing maximum output power.
It is also instructive to examine calculated heat flow through the leg as this helps to illustrate thermal‐to‐electrical conversion process. Figure 22 shows heat flow corresponding to temperature profiles depicted in Figure 21. From the hot side to the cold side of the leg (or right to left on the plot), heat flow is reduced as thermal energy is converted to electrical power and delivered to the load. Examining the specific case of
This discussion has highlighted a simple, but powerful temperature‐dependent phenomenological model for precisely calculating temperature profiles, heat flows, power outputs, and efficiencies in a single leg of TEG. Full TEG device modeling is accomplished by simultaneously solving the discrete heat balances as described for each leg (subject to hot side and cold side boundary conditions) along with the simultaneous energy balance relationship required for electrical power being delivered to the load.
Figure 24 shows calculated efficiency using the
Electrical and thermal contact resistivities are defined to be zero, but could easily be included as finite quantities, which would add penalties to the efficiency. Slope of efficiency identified in the best‐fit is quantified first using unitless efficiency data. The equation is then re‐included on the plot after converting to percent. This is so that ZTmaximum can be calculated using
Therefore, to confirm measurements for device fabricated using materials, from which measurements were collected, it could be assembled and the efficiency is measured. If measurements are accurate and not overestimated, then performance should be consistent with ZTmaximum value equals to 0.64. The slope of the data should be roughly 0.0002/K.
In conclusion, we have presented several novel approaches to the significant challenge of accurately determining the thermal conductivity of thermoelectric materials. The new solutions can be much faster experimentally, and they successfully address several sources of experimental error. The overall result is significantly reduced error, which may reduce uncertainty by a factor of 2 or more. Further, we introduce new approaches to compare device performance with physical property measurements as a novel means of confirming measurements. Using this approach, the new measurements can be clearly seen to yield physical property measurements which are more consistent with physical device performance.
The first new thermal conductivity measurement method,
The truest test of the accuracy of measurement is comparison with fabricated devices. To support the validation of measurements of individual material properties, we have outlined new device metrics, which allows comparison between theoretical and measured device efficiency. We outline a new