Measurements of Thermoelectric Properties of Identical Bi-Sb Sample in Magnetic Fields and Influence of Sample Geometry

Abstract

The influence of sample geometry on various measured physical properties (including the magneto-Seebeck effect, Nernst effect, magnetoresistance effect, Hall effect, and thermal conductivity) in the presence of a magnetic field was examined using a polycrystalline Bi-Sb sample. The sample, consisting of polycrystalline Bi₈₈Sb₁₂, was prepared through spark plasma sintering and subsequent annealing. Measurements of the physical properties were conducted under a magnetic field of 5 T, and the obtained values were compared with simulated results derived using the finite element method for different sample geometries. Distinct shapes were found to be necessary for accurate measurements, with each physical property requiring a specific aspect ratio of sample length (l) to width (w). These ratios were determined to be l/w > 0.57, 2.9, 4.2, 1.2, and 3.1 for the magnetoresistance, Hall, two-wire magneto-Seebeck, four-wire magneto-Seebeck, and Nernst effects, respectively. Additionally, to achieve a minimal error of less than 2% in thermal conductivity measurement, a thermal conductance ratio of K_s/K_w > 27 was necessary, where K_s and K_w denote the thermal conductance of the sample and lead wire, respectively.

 

This Paper was Originally Published in Japanese in J. Thermoelec. Soc. Jpn. 20 (2023) 67–74. The vertical axis label of the graph in Fig. 9 is modified.

1. Introduction

Bi-Sb alloys are recognized for their high dimensionless figure of merit, zT, particularly at temperatures below room temperature. Among these alloys, Bi₈₈Sb₁₂ demonstrates the highest zT value of 0.55 at 120 K in the absence of a magnetic field. However, the application of an external magnetic field leads to an increase in zT. For instance, a zT value of 1.28 at 220 K was reported under a 1 T magnetic field [1]. Additionally, even at room temperature (300 K), a zT value of 0.84 has been observed in the presence of a magnetic field, indicating notable performance. The introduction of an external magnetic field to Bi-Sb alloys induces changes in three physical properties: Seebeck thermopower (S), electrical resistivity (ρ), and thermal conductivity (κ), thereby enhancing zT. This phenomenon, where the Seebeck thermopower changes in response to a magnetic field, is termed the magneto-Seebeck effect. Conversely, Bi-Sb alloys also exhibit a significant Nernst effect, which is a thermoelectric phenomenon observed in the presence of a magnetic field [2]. The Nernst effect entails the induction of a thermoelectromotive force perpendicular to both the temperature gradient and magnetic field. This property simplifies device structures due to the resulting voltage obtained in the direction perpendicular to the temperature difference [3]. Particularly noteworthy are devices utilizing the anomalous Nernst effect (ANE), which occurs in magnetic materials. These devices are deemed suitable for mass production as they can operate without requiring an external magnetic field [4]. Recent research and development efforts have been directed towards applying the ANE to heat flux sensors [5].

Our research group has dedicated efforts to exploring the thermoelectric effect of Bi-Sb alloys under the influence of a magnetic field, with the aim of enhancing their thermoelectric performance. In pursuit of this goal, we have undertaken material development, device fabrication, and the establishment of characteristic evaluation techniques [6–8]. When conducting measurements of various physical properties in the presence of a magnetic field, the observed values are significantly influenced by the sample geometry, contrasting with measurements conducted in the absence of a magnetic field [9–21]. Typically, for accurate measurement of the magneto-Seebeck, Nernst, magnetoresistance, and Hall effects, a rod-like shape with a high aspect ratio (l/w) length, l, and width, w is preferred. In the presence of a magnetic field, a perpendicular potential difference to the temperature gradient (current) arises due to the Nernst effect (Hall effect). However, this potential difference diminishes when the equipotential surface near the end becomes parallel to the end surface, as the metal electrode at the sample end becomes shorted. Generally, to ensure proper evaluation of the Nernst and Hall effects, the aspect ratio (l/w) should be equal to or greater than 3–4 [12, 20]. Conversely, when measuring thermal conductivity, employing a flat sample with a sizable cross-sectional area is advantageous to minimize heat leakage from the sample to the surroundings [22]. This requirement differs from the sample geometry needed for measurements of properties such as the magneto-Seebeck and Nernst effects. In our research endeavors, we have utilized bulk samples to assess the physical properties of materials under a magnetic field (including the magneto-Seebeck effect, Nernst effect, magnetoresistance effect, Hall effect, and thermal conductivity) using identical samples. A standard sample measuring 3 × 3 × 12 mm³ has been adopted due to its aspect ratio of approximately 4, adequate cross-sectional area, and ease of fabrication of the sintered body [6, 7].

In this study, we aimed to ascertain the dimensionless figures of merit using a consistent sample. To achieve this, we explored the requisite sample geometries for accurately measuring the Seebeck and Nernst thermopowers, diagonal and Hall resistivities, and thermal conductivity in the presence of a magnetic field. Utilizing a Bi-Sb alloy fabricated through spark plasma sintering (SPS), we conducted measurements of various physical properties under a magnetic field. Subsequently, based on the acquired physical properties, we simulated the measured values using the finite element method for different sample geometries.

2. Experimental

2.1 Sample preparation

Bi₈₈Sb₁₂, recognized for its substantial zT value attributed to the magneto-Seebeck effect, was chosen as the material for simulating the evaluation of physical properties. Single crystals of Bi-Sb alloys exhibit pronounced anisotropy in their physical properties. However, to streamline the simulation process by disregarding this anisotropy, we fabricated a sintered body with minimal anisotropy [6]. Bi and Sb shots (5N grade, Furuuchi Chemical) were individually ground using an agate mortar and pestle, and the resulting powders were blended to achieve the desired atomic composition ratio. Subsequently, the mixture underwent sintering using the SPS method (SPS-515S, Fuji Electronic Industrial) at 220°C and 50 MPa for 10 minutes, resulting in the formation of a disc-shaped ingot with a diameter of 15 mm and a thickness of approximately 3 mm. To facilitate alloying through solid-phase reaction, the produced ingot was vacuum-sealed in a quartz ampoule and annealed at 250°C for one week in an electric furnace. The resulting ingot was then shaped to dimensions of 2.909 × 3.405 × 11.074 mm³ through wire sawing and polishing.

2.2 Sample measurement setup

Figure 1 depicts both a schematic (a) and a photograph (b) of the measurement setup featuring the fabricated Bi₈₈Sb₁₂ bulk material. To establish a temperature gradient along the longitudinal axis of the rectangular sample, copper plates with a thickness of approximately 0.4 mm were affixed to both ends of the sample using silver epoxy (H 9683, Namics) to act as heat baths on the cold and hot sides. Copper wires, 25 µm in diameter, were bonded to the upper and lower copper plates with silver epoxy. These wires served to apply an electrical current for assessing the galvanomagnetic effect and to measure the electromotive force for evaluating the thermomagnetic effect. Three copper wires with diameter of 25 µm were attached to one side of the sample using silver epoxy, while one wire was attached to the opposite side to measure the voltages corresponding to each effect. The three copper wires on one side were positioned at 1/4, 1/2, and 3/4 of the sample’s height, while another wire was situated at the center of the opposite side. Due to manual installation of the thin metal wires, their actual positions deviated from these designated positions; Fig. 1(a) displays the distances between each electrode measured using a digital microscope. Reducing the diameter of the contact point when connecting the thin metal wire to the sample can mitigate measurement errors. For instance, in the case of diagonal resistance, the distance between the electrodes used for measurement is approximately 6 mm. Thus, to limit the error within 1%, the contact point’s diameter should ideally be ≤60 µm. However, reducing the contact-point diameter is challenging when manually connecting thin metal wires using silver epoxy. The contact-point diameter of the sample utilized in the experiment ranged from approximately 0.3 to 0.4 mm. Consequently, the error attributable to the contact point diameter in this study may be as high as 7%.

Fig. 1

(a) Schematic and (b) photograph of set-up for measurements of diagonal and Hall resistivity, Seebeck and Nernst thermopower, and thermal conductivity for Bi₈₈Sb₁₂ bulk.

A compact 120-Ω heater (SGD-1.5/120-LY11, Omega Engineering) was affixed to the hot-side copper plate using silicone adhesive (1225B, ThreeBond) to induce a temperature differential for measuring the Seebeck and Nernst thermopowers, as well as thermal conductivity. To minimize heat dissipation from the heater, two manganin wires (25 µm in diameter), possessing low thermal conductivity, were attached to each positive and negative conductor wire of the heater. For temperature differential measurement, a T-type differential thermocouple with a 25 µm diameter was thermally bonded to both copper plates, utilizing silver epoxy on the cold side and silicone adhesive on the hot side. The cold-side contact point of the thermocouple was insulated with silicone adhesive to prevent the formation of a closed circuit with the sample. Subsequently, the fabricated sample slated for measurement was secured to the AlN substrate of the sample holder using silicone adhesive.

2.3 Measurement methods

A Cernox temperature sensor (Lakeshore) was integrated into the sample holder to gauge the absolute temperature of the cold side. The sample holder was installed within a Gifford-McMahon (GM) refrigerator-type cryostat, with fiber reinforced plastics (FRP) sheets strategically placed between the refrigerator and sample holder. Temperature control was facilitated using a heater [22, 23], effectively suppressing temperature fluctuations within the sample holder to the milli-Kelvin range. Measurements of diagonal/Hall resistivities, Seebeck/Nernst thermopowers, and thermal conductivity in a magnetic field at 300 K were conducted under a vacuum atmosphere of <10⁻³ Pa.

For the diagonal and Hall resistance (galvanomagnetic effect) measurements, an alternating current of 1.0 mA and 1.234 Hz was applied using a current source (6221, Keithley) (see Fig. 2(a)). The magnetic field dependence of the two resistances was simultaneously assessed using two lock-in amplifiers (SR860, Stanford Research Systems). Seebeck and Nernst thermopowers (thermomagnetic effect) along with thermal conductivity were measured concurrently utilizing the steady-state method (see Fig. 2(b)). Initially, the resistance of the upper heater was determined at the measured temperature using the four-wire method. To induce a temperature gradient, a current source (6221, Keithley) was employed to deliver current to the heater via two parallel manganin wires, minimizing heat generation in the wires. The heat passing through the sample was quantified based on the heater resistance (previously measured) and the applied current. Since the resistance of the two parallel thin manganin wires accounted for approximately 8% of the heater resistance, roughly 4% of the heat generated by the heater (equivalent to half of the heat generated by the thin manganin wires) might flow into the sample. The temperature difference across the samples was determined from the electromotive force of the differential thermocouple, measured using a nanovoltmeter (2182A, Keithley), and the absolute temperature on the cold side. To nullify offset voltages, the voltage of each electrode was assessed at two temperature differences (e.g., 0.2 K and 1.0 K at 0 T) utilizing a digital multimeter (2002, Keithley). An external magnetic field, generated using a superconducting coil (Cryogen Free Magnet System, Cryogenic), was swept between ±5 T at a rate of 0.17 T min⁻¹, aligned perpendicular to the longitudinal direction of the sample.

Fig. 2

Schematics of measurements for (a) galvanomagnetic effect and (b) thermomagnetic effect and thermal conductivity.

3. Measurement Results

3.1 Galvanomagnetic effect

To assess the galvanomagnetic effect, two voltages were recorded to ascertain the diagonal and Hall resistivities: V_l along the longitudinal direction and V_t across the transverse direction of the sample. Utilizing the acquired V_l and V_t values alongside the applied current, I, resistivities ρ_l and ρ_t were calculated, factoring in the sample dimensions and the spacing between the electrodes, using the following expressions:

  

\begin{equation} \rho_{\text{l}} = \frac{V_{\text{l}}}{I}\frac{wd}{g}, \end{equation} (1)

  

\begin{equation} \rho_{\text{t}} = \frac{V_{\text{t}}}{I}d, \end{equation} (2)

where w, d, and g represent the width and depth of the sample and the spacing between the electrodes, respectively. These recorded resistivities comprise a blend of the diagonal resistivity, ρ_d, and the Hall resistivity, ρ_H, due to the displacement of the electrode positions. Considering ρ_d as an even function with respect to the magnetic field, B, based on ρ_l, and ρ_H as an odd function with respect to B based on ρ_t, ρ_d and ρ_t are determined as follows:

  

\begin{equation} \rho_{\text{d}} = \frac{\rho_{\text{l}}(B) + \rho_{\text{l}}(-B)}{2}, \end{equation} (3)

  

\begin{equation} \rho_{\text{H}} = \frac{\rho_{\text{t}}(B) - \rho_{\text{t}}(-B)}{2}. \end{equation} (4)

Figure 3(a) illustrates the magnetic field dependency of ρ_d and ρ_H up to 5 T, measured at 300 K. In the absence of a magnetic field, ρ_d = 3.48 µΩ m, which escalates to 8.58 µΩ m at 5 T due to the magnetoresistance effect; ρ_H = −2.60 µΩ m at 5 T.

Fig. 3

Measured magnetic field dependences of (a) diagonal and Hall resistivity, (b) Seebeck and Nernst thermopowers, (c) thermal conductivity, and (d) dimensionless figure-of-merit for Seebeck and Nernst effects.

3.2 Thermomagnetic effect

Concerning the thermomagnetic effect, for assessing the magneto-Seebeck and Nernst thermopowers, two potential differences were gauged: ΔV_l along the longitudinal direction and ΔV_t across the transverse direction of the sample. From the acquired ΔV_l and ΔV_t values at temperature disparities ΔT₁ and ΔT₂, thermopowers α_l and α_t were computed, taking into account the sample dimensions, as follows:

  

\begin{equation} \alpha_{\text{l}} = \frac{\Delta V_{\text{l,2}} - \Delta V_{\text{l,1}}}{\Delta T_{2} - \Delta T_{1}}, \end{equation} (5)

  

\begin{equation} \alpha_{\text{t}} = \frac{\Delta V_{\text{t,2}} - \Delta V_{\text{t,1}}}{\Delta T_{2} - \Delta T_{1}}\frac{l}{w}, \end{equation} (6)

where l represents the sample length. These recorded thermopowers encompass a fusion of the magneto-Seebeck thermopower, S_m, and the Nernst thermopower, NB, due to the shift in electrode positions. Hence, akin to the galvanomagnetic effect, by regarding S_m as an even function relative to B based on a_l, and NB as an odd function relative to B based on a_t, a_d and a_t are assessed as:

  

\begin{equation} S_{\text{m}} = \frac{\alpha_{\text{l}}(B) + \alpha_{\text{l}}(-B)}{2}, \end{equation} (7)

  

\begin{equation} \mathit{NB} = \frac{\alpha_{\text{t}}(B) - \alpha_{\text{t}}(-B)}{2}. \end{equation} (8)

It is worth noting that the magneto-Seebeck effect displays asymmetry concerning the magnetic field due to the Umkehr effect in the case of a single crystal [24]. However, for this study, we disregarded this effect since the sample utilized was a sintered body with low anisotropy [6]. Figure 3(b) illustrates the magnetic field dependence of S_m and NB up to 5 T, measured at 300 K. In the absence of a magnetic field, S_m = −80.1 µV K⁻¹, which escalates to −118 µV K⁻¹ at 5 T due to the magneto-Seebeck effect; NB = 63.2 µV K⁻¹ at 5 T.

3.3 Thermal conductivity

The magnetic field dependence of the thermal conductivity was determined by considering the amount of power input into the heater, Q, and the sample dimensions, as:

  

\begin{equation} \kappa_{\text{l}} = \frac{Q_{2} - Q_{1}}{\Delta T_{2} - \Delta T_{1}}\frac{l}{wd}. \end{equation} (9)

Considering the thermal conductivity, κ_d, as an even function with respect to B based on κ_l, κ_d is calculated as:

  

\begin{equation} \kappa_{\text{d}} = \frac{\kappa_{\text{l}}(B) + \kappa_{\text{l}}(-B)}{2}. \end{equation} (10)

Figure 3(c) shows the magnetic field dependence of κ_d up to 5 T measured at 300 K. In the absence of a magnetic field, κ_d = 3.96 W m⁻¹ K⁻¹, which decreases to 3.25 W m⁻¹ K⁻¹ at 5 T.

3.4 Dimensionless figure of merit

If the anisotropy of the physical properties is ignored, the dimensionless figures of merit, z_ST and z_NT, derived from the Seebeck and Nernst effects are expressed as

  

\begin{equation} z_{\text{S}}T = \frac{S_{\text{m}}{}^{2}}{\rho_{\text{d}}\kappa_{\text{d}}}T, \end{equation} (11)

  

\begin{equation} z_{\text{N}}T = \frac{(\mathit{NB})^{2}}{\rho_{\text{d}}\kappa_{\text{d}}}T. \end{equation} (12)

Here, T denotes the absolute temperature of the sample. Figure 3(d) presents the magnetic field dependence of z_ST and z_NT computed from each physical property measured at 300 K. In the absence of a magnetic field, z_ST = 0.14, which rises to 0.16 at approximately 1.5 T, then gradually diminishes with increasing magnetic field up to 5 T. Conversely, z_NT steadily increases with the magnetic field’s escalation, reaching 0.043 at 5 T.

3.5 Relationship between the measured values and physical properties

The thermoelectric effect in a magnetic field is expressed based on the electrical conductivity, σ; Seebeck coefficient, S; Hall coefficient, R; Nernst coefficient, N; thermal conductivity, κ; Righi–Leduc coefficient, L; electric field, E; current density, j; energy flux density, q; and electrochemical potential, φ, and is as follows [25]:

  

\begin{equation} \mathbf{E} = \frac{1}{\sigma}\mathbf{j} + S\nabla T + R\mathbf{B} \times \mathbf{j} + N\mathbf{B} \times \nabla T \equiv \underline{\boldsymbol{\rho}}\mathbf{j} + \underline{\boldsymbol{\alpha}} \nabla T, \end{equation} (13)

  

\begin{align} \mathbf{q} - \phi \mathbf{j} &= ST\mathbf{j} - \kappa \nabla T + N\mathbf{B} \times \mathbf{j} + L\mathbf{B} \times \nabla T \\ &\equiv \underline{\boldsymbol{\alpha}} T\mathbf{j} - \underline{\mathbf{K}} \nabla T, \end{align} (14)

The electrical resistivity tensor ρ, thermoelectric power tensor α, and thermal conductivity tensor K are introduced to the right sides of eqs. (13) and (14); these parameters express their effects in the magnetic field as second-order tensors. Under isothermal conditions (∂T/∂y = 0), when a magnetic field and current are applied along the [001] and [100] directions, respectively, ρ_xx represents the (isothermal) diagonal resistivity, and ρ_yx denotes the (isothermal) Hall resistivity. Similarly, when a magnetic field and temperature gradient are applied in the [001] and [100] directions, respectively, α_xx denotes the (isothermal) Seebeck thermopower, and α_yx represents the (isothermal) Nernst thermopower. The diagonal resistivity, ρ_d, and Hall resistivity, ρ_H, were measured under isothermal conditions due to the application of the AC method [26] and correspond to ρ_xx and ρ_yx, respectively. Conversely, the measured magneto-Seebeck thermopower, S_m, and Nernst thermopower, NB, were derived from the (adiabatic) Seebeck and (adiabatic) Nernst effects, respectively, owing to the utilization of the steady-state method and thermally insulated sample side [26]. The measured S_m and NB can be expressed using α_xx and α_yx under isothermal conditions as follows [27]:

  

\begin{equation} S_{\text{m}} = \alpha_{xx} + \alpha_{yx}\frac{\nabla_{y}T}{\nabla_{x}T}, \end{equation} (15)

  

\begin{equation} \mathit{NB} = \alpha_{yx} + \alpha_{xx}\frac{\nabla_{y}T}{\nabla_{x}T}, \end{equation} (16)

where the difference in thermopower between the measurement and isothermal conditions is due to the Righi–Leduc effect, ∇_yT = (K_yx/K_xx)∇_xT, which causes heat flow perpendicular to the temperature gradient and magnetic field [26]. Note that K_yx/K_xx is small and thus generally ignored for Bi-Sb alloys [28].

4. Discussion

4.1 Effect of sample geometry

The observed galvanomagnetic and thermomagnetic effects are significantly influenced by the sample geometry [9–21]. In particular, the attachment of a metal electrode to the sample end aligns the equipotential surface near the end parallel to the end surface, rendering it susceptible to transverse effects such as the Hall and Nernst effects [10, 12, 20]. If the aspect ratio, l/w, of the sample is small, the influence of the end electrodes also extends to the center of the sample, where the transverse effect is measured; hence, a large l/w is generally preferred in transverse effect measurements. Furthermore, in the magneto-Seebeck effect measurement using the two-wire method, a short-circuit current is induced near the end electrodes due to the Nernst effect [20], consequently leading to the Hall effect and affecting the measured values. However, in thermal conductivity measurements, the use of a flat sample with a large cross-sectional area, facilitating heat flow to the sample, is favored to mitigate the effects of heat leakage caused by factors such as wiring and radiation [21]. This differs from the sample geometry required for galvanomagnetic and thermomagnetic effect measurements. Nonetheless, to evaluate the figure of merit of a material, measuring all physical properties of identical samples is desirable. Moreover, to assess each physical property with minimal error, specific conditions regarding sample geometry are applicable. Therefore, we conducted a finite element simulation using COMSOL Multiphysics 4.4 to investigate the influence of sample geometry on the measured values of each physical property. The physical property values used in the simulation were the measurement results for the Bi₈₈Sb₁₂ sintered body obtained at 300 K and 5 T. The galvanomagnetic/thermomagnetic effects and thermal conductivity were simulated separately due to their differing configurations for geometric dependency.

4.2 Galvanomagnetic and thermomagnetic effects

To explore the aspect ratio’s influence on the measured galvanomagnetic effect (magnetoresistance and Hall effects) and thermomagnetic effect (magneto-Seebeck and Nernst effects), we maintained the sample’s cross-sectional area at 3 × 3 mm² while varying the sample length from 1 mm to 20 mm. Figure 5 illustrates the simulation configuration for a sample length of 12 mm as an example. We assumed a setup where copper plates with a thickness of 0.4 mm and the same cross-sectional area as the rectangular parallelepiped Bi-Sb bulk were directly affixed to both ends of the sample to simplify the simulation configuration, excluding the thin metal wires typically attached to the actual sample for measurement. The potential probe positions were designated on one side at 1/4, 1/2, and 3/4 of the sample’s height, with another wire positioned at the center of the opposite side. For the simulation of the galvanomagnetic effect, a 1-mA current was applied from the top surface of the upper copper plate to the bottom surface of the lower copper plate. Subsequently, similar to the experimental setup, ρ_d and ρ_H were deduced from the potentials corresponding to each probe, sample dimensions, and symmetry concerning the magnetic field. In the simulation of the thermomagnetic effect, the upper copper plate’s top surface was set at 301 K, and the bottom surface of the lower copper plate was set at 300 K. Then, akin to the experiment, S_m and NB were determined from the potentials corresponding to each probe, sample dimensions, and symmetry regarding the magnetic field. While the experiment measured the Seebeck thermopower using the two-wire method, the simulation also included the Seebeck thermopower of the two-wire method, derived from the potentials and temperatures at the copper plates. Additionally, the Seebeck thermopower measured by the four-wire method was determined from the potentials and temperatures at the probes on the sample’s side.

Fig. 4

Influence of electrodes on measured magneto-Seebeck thermopower using two-wire method.

Fig. 5

Configuration of FEM simulation for galvano- and thermo-magnetic effects (e.g., 12 mm length).

The sample length dependency of the measured galvanomagnetic and thermomagnetic effects obtained from the simulations and the input physical properties are illustrated in Fig. 6. In Fig. 6, the left axis displays the simulated and input values, while the right axis depicts the error in the simulated value relative to the input value. Each data point is interpolated using a cubic spline. As depicted in Fig. 6(a), ρ_d = 8.82 µΩ m for a sample length of 1 mm (l/w = 0.33), exhibiting a 2.8% increase compared to the input value of 8.58 µΩ m. This discrepancy arises due to the amplified distortion of the current line for a small aspect ratio [11, 14]. As the aspect ratio rises, the region where the current flows parallel to the longitudinal direction of the sample expands, leading to the convergence of the measured value towards the input value. To keep the geometry-related error within 2% or 1% of the input value, the length must be ≥1.6 mm (l/w = 0.53) or ≥2.9 mm (l/w = 0.97), respectively. Figure 6(b) illustrates that ρ_H = −0.489 µΩ m for a sample length of 1 mm (l/w = 0.33), with the absolute value being 81% smaller than the input value of −2.60 µΩ m. This occurs because although a potential difference arises in the transverse direction due to the Hall effect, the short-circuit at the end electrode aligns the equipotential surface parallel to the end surface, reducing the transverse potential difference, even at the sample center. As the aspect ratio increases, the influence of the end electrodes diminishes at the center, and the simulated value converges towards the input value. To limit the geometry-related error within 2% or 1% of the input value, the length must be ≥8.6 mm (l/w = 2.9) or ≥10 mm (l/w = 3.3), respectively.

Fig. 6

Sample length dependences of simulated measurement values of galvano- and thermo-magnetic effects.

As depicted in Fig. 6(c), when simulated using the two-wire method, S_m = −133 µV K⁻¹ for a sample length of 1 mm (l/w = 0.33), indicating a 12% increase compared to the input value of −118 µV K⁻¹. This discrepancy arises due to the significant effect of the short-circuit current at the end electrode, as illustrated in Fig. 4, particularly noticeable when the aspect ratio is small. Reversing the sign of the input value of the Hall coefficient results in a smaller simulated |S_m| than the absolute value of the input S_m, indicating that the substantial increase in simulated |S_m| is attributable to the Hall effect. As the aspect ratio increases, the influence of the end electrodes diminishes at the center, converging the result towards the input value. To keep the geometry-related error within 2% of the input value, the length must be ≥13.5 mm (l/w = 4.5); however, even with a length of 20 mm, the error will not fall within 1%. In the four-wire method, where the influence of the end electrodes is reduced, S_m = −124 µV K⁻¹ for a sample length of 1 mm (l/w = 0.33), with the geometry-related error at 4.3% relative to the input value. As the aspect ratio increases, the geometry-related error falls within 2% or 1% for lengths ≥2.67 mm (l/w = 0.89) or ≥3.40 mm (l/w = 1.13), respectively. Figure 6(d) illustrates that NB = 11.5 µV K⁻¹ for a sample length of 1 mm (l/w = 0.33), which is 82% smaller than the input value of 63.2 µV K⁻¹. This discrepancy arises because, despite a potential difference occurring in the transverse direction due to the Nernst effect, a short-circuit at the end electrode aligns the equipotential surface parallel to the end surface, diminishing the transverse potential difference at the center of the sample. As the aspect ratio increases, the influence of the end electrodes diminishes at the center, bringing the simulated value closer to the input value. To maintain the geometry-related error within 2% or 1% of the input value, the length must be ≥9.5 mm (l/w = 3.1) or ≥11.9 mm (l/w = 4.0), respectively.

4.3 Thermal conductivity

The simulation of thermal conductivity differs from that of the galvanomagnetic and thermomagnetic effects. To account for heat leakage from the connected thin metal wires, we adopted a configuration that included these wires (Fig. 7). Three copper wires with a diameter of 25 µm were connected on one side at 1/4, 1/2, and 3/4 of the height of the sample, with another wire placed at the center of the opposite side. Similarly, as in the experiment, two copper wires, one constantan wire, and four manganin wires with a diameter of 25 µm were placed in the hot-side heat bath. The length of each wire was assumed to be the shortest for the connection on the cold side. While heat leakage in reality also occurs due to thermal radiation, accurately incorporating the emissivity of actual measurements is challenging. Therefore, radiation was not considered in this simulation, and only heat leakage caused by thermal conduction from the thin metal wires was included. As the lengths of the sample and thin metal wires are proportional, the ratio of heat leakage from the thin metal wires remains constant. Consequently, the simulated thermal conductivity is not strongly dependent on the sample length. Therefore, in the thermal conductivity simulation, the sample length was fixed at 12 mm, while the sample cross-sectional area was varied from 1 × 1 mm to 5 × 5 mm² to determine the aspect ratio dependence. The temperatures at the top surface of the upper copper plate and the bottom surface of the lower copper plate were set at 301 K and 300 K, respectively, with the end of each thin metal wire fixed at 300 K. Subsequently, the thermal conductivity was determined based on the heat flow from the top surface of the upper copper plate and the sample dimensions. Additionally, to simplify the simulation, only the diagonal resistivity, Seebeck thermopower, and thermal conductivity in the magnetic field were considered, while the Hall, Nernst, and Righi–Leduc effects were ignored.

Fig. 7

Configuration of FEM simulation for thermal conductivity (e.g., 3 mm × 3 mm cross-sectional area).

In Fig. 8, the left axis illustrates the shape dependence of the simulated thermal conductivity alongside the input values, while the right axis represents the error relative to the input value. For a sample cross-sectional area of 1 mm² (sample width: 1 mm), κ_d = 3.98 W m⁻¹ K⁻¹, which exceeds the input value of 3.25 W m⁻¹ K⁻¹ by 22%. This discrepancy arises because, with a small sample cross-sectional area, the proportion of heat leakage from the thin metal wires to that passing through the sample is high. As the cross-sectional area of the sample increases, the influence of heat leakage from the thin metal wires diminishes, and the value approaches the input value. To keep the error attributed to heat leakage from the thin metal wires within 2% or 1% of the input value, the cross-sectional area must be ≥10.8 mm² (w = 3.28 mm) or ≥22.1 mm² (w = 4.70 mm), respectively.

Fig. 8

Sample geometry dependences of simulated measurement values of thermal conductivity.

4.4 Dimensionless figure of merit

A larger aspect ratio is preferable for measuring the galvanomagnetic and thermomagnetic effects, while a larger cross-sectional area is preferable for measuring thermal conductivity. Hence, selecting a sample shape that matches the measurement can reduce errors if each physical property value is measured in separate samples to evaluate the dimensionless figure of merit, zT, of a material. However, when zT is determined from physical properties measured using different samples, caution is necessary as the properties can vary among the samples and exhibit anisotropy. Although measuring all physical properties in the same sample is ideal, as discussed earlier, the sample geometry is constrained to ensure measurement of each physical property with minimal errors. Figure 9 illustrates the ranges of error (1%, 2%, 3%, 5%, and 10%) for each physical property due to sample geometry, as obtained from simulations when determining zT for the magneto-Seebeck effect using the two-wire method and the Nernst effect. To consider the sample geometry’s influence in a dimensionless manner, the horizontal axis represents the ratio of the sample’s thermal conductance, K_s, to that of all thin metal wires, K_w (K_s/K_w), while the vertical axis represents the sample’s aspect ratio. The Bi₈₈Sb₁₂ sample used in this measurement is denoted by an asterisk. The error range differs between the magneto-Seebeck effect (two-wire) and the Nernst effect: a larger aspect ratio is required for the magneto-Seebeck effect compared to the Nernst effect when measured using the two-wire method. For the Bi₈₈Sb₁₂ sample shape used in this measurement, the error in NB is within 2%, while that in S_m is within 3%. Conversely, κ_d may exhibit an error of up to 3% due to heat leakage from all thin metal wires.

Fig. 9

Range of sample geometry and thermal conductance conditions, with each measured error of property value <1, 2, 3, 5, and 10%.

Thus, to measure each physical property with minimal errors in a magnetic field, certain conditions are imposed on the sample geometry, and the size of the fabricated sample and its installation location in the measuring device are also restricted. Therefore, selecting a specific sample shape is not always feasible. In the case of the physical properties of the Bi₈₈Sb₁₂ sintered body used in this experiment, the errors in each physical property due to the sample geometry could be kept within 2–3% using typical sample dimensions of 3 × 3 × 12 mm³. However, in actual measurements, errors including a maximum error of approximately 7% in ρ_d due to the contact point diameter of the electrode, an error of approximately 4% in κ_d due to heat inflow resulting from heat generation in the thin manganin wires, and an error in κ_d due to heat leakage owing to thermal radiation, can occur.

5. Conclusion

In this study, we investigated the influence of sample geometry on various physical properties measured in a magnetic field, including the magneto-Seebeck effect, Nernst effect, magnetoresistance effect, Hall effect, and thermal conductivity. A bulk Bi₈₈Sb₁₂ sample was prepared using spark plasma sintering (SPS) and annealing, and its magnetic field dependence was measured up to 5 T at 300 K. We then simulated the influence of sample geometry on the measured values of each physical property using the finite element method with the experimentally obtained physical properties of the Bi₈₈Sb₁₂ bulk sample. Consequently, we found that the appropriate sample geometry varied for each physical property. To reduce the geometry-related error to less than 2%, the aspect ratio l/w must be >0.57, 2.9, 4.2, 1.2, and 3.2 for the magnetoresistance, Hall, two-wire magneto-Seebeck, four-wire magneto-Seebeck, and Nernst effects, respectively. However, for thermal conductivity, a ratio of K_s/K_w > 27 is necessary, considering only heat leakage from the thin metal wires. As previously discussed, when evaluating the zT of the thermoelectric effect in a magnetic field using a single sintered Bi₈₈Sb₁₂ sample, we identified the appropriate sample geometry to minimize the measurement error for each physical property.

Acknowledgments

This study is based on results from a project commissioned by the Mitou Challenge 2050 of the Advanced Research Program for Energy and Environmental Technologies of NEDO. Support was provided in part by JSPS KAKENHI Early-Career Scientists (Grant Number 19K15297), JSPS KAKENHI Grant-in-Aid for Scientific Research (C) (Grant Number 22K04232), JSPS KAKENHI Fostering Joint International Research (B) (Grant Number 18KK0132), JST ACT-X (Grant Number JPMJAX21KI), and JST Mirai-Program (Grant Number JPMJMI19A1).

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