Scalable Nernst Thermoelectric Power using a Coiled Galfenol Wire

The Nernst thermopower usually is considered far too weak in most metals for waste heat recovery. However, its transverse orientation gives it an advantage over the Seebeck effect on non-flat surfaces. Here, we experimentally demonstrate the scalable generation of a Nernst voltage in an air-cooled metal wire coiled around a hot cylinder. In this geometry, a radial temperature gradient generates an azimuthal electric field in the coil. A Galfenol (Fe$_{0.85}$Ga$_{0.15}$) wire is wrapped around a cartridge heater, and the voltage drop across the wire is measured as a function of axial magnetic field. As expected, the Nernst voltage scales linearly with the length of the wire. Based on heat conduction and fluid dynamic equations, finite-element method is used to calculate the temperature gradient across the Galfenol wire and determine the Nernst coefficient. A giant Nernst coefficient of -2.6 ${\mu}$V/KT at room temperature is estimated, in agreement with measurements on bulk Galfenol. We expect that the giant Nernst effect in Galfenol arises from its magnetostriction, presumably through enhanced magnon-phonon coupling. Our results demonstrate the feasibility of a transverse thermoelectric generator capable of scalable output power from non-flat heat sources.

Thermoelectric power generation is considered an environmentally friendly approach to convert waste heat into electrical energy. The conventional way is to take advantage of the Seebeck effect, where an electric field is generated longitudinally along a thermal gradient. On the other hand, a transverse thermoelectric effect, called the Nernst effect, generates an electric field perpendicular to the plane between an applied temperature gradient and magnetic field. Despite the weaker nature of the Nernst effect compared to the Seebeck effect, the Nernst effect offers several advantages and flexibilities for thermopower generation [1]. More specifically, due to the orthogonality of the Nernst electric field to the thermal gradient, the Nernst thermopower can be increased linearly with the thermoelectric generator length without introducing the conventional thermopile structure (n-and p-type semiconductor structures connected in series). This is specifically useful for cylindrical heating geometries, where a device originally proposed by Norwood [2] can be implemented. This device consists of a wire wrapped around a hot cylinder to produce a Nernst voltage that scales linearly with the wire length, i.e. the number of turns. As pointed out by Norwood [2], "the value of a device of this type is that the heat pumping material has no metal-semiconductor junctions and there are no heat leakage paths (neglecting end effects) except through the material." The scalable Nernst thermopower generation possible in the coiled wire geometry also provides a more easily manufactured and economical approach than the conventional Seebeck thermoelectric generator. Thus, the Nernst effect offers an attractive alternative for thermoelectric waste heat recovery on non-flat surfaces, such as hot exhaust pipes.
In magnetic materials, the total Nernst effect consists of ordinary ( �⃗ ) and anomalous ( �⃗ ) components: [2] where and are the ordinary and anomalous Nernst coefficients respectively, 0 is the vacuum permeability, � �⃗ is the applied field, and ��⃗ is the magnetization. Sakuraba recently evaluated the potential performance of a thermoelectric generator in a coil geometry and showed that in order to reach practical power densities useful for waste heat recovery, larger Nernst coefficients are necessary compared to those typically found in metals [1]. To that end, phonon drag had been shown to enhance the Nernst coefficient at low temperatures in certain material systems [3][4]. More recently, Watzman et al. reported that the anomalous Nernst coefficient also benefits from magnon drag as shown in single crystal Fe where is dominated by magnon drag up to 200 K [5]. In this study, we propose using a magnetostrictive metallic metal, Galfenol (Fe100-xGax), as a pathway to increase the power factor due to its enhanced Nernst coefficient from phonon and magnon coupling, as well as its large intrinsic electrical conductivity. We demonstrate Nernst voltage generation in a coiled Galfenol wire, characterize its Nernst coefficient, and offer an experimental proof-of-concept for a novel and scalable thermoelectric generator ideal for cylindrical heaters.
Galfenol (FeGa) represents a category of mechanically robust, high strength, and low cost magnetostrictive alloys. The magnetostriction is shown to be as high as ~ 886 ppm for Fe83Ga17 with Tb doping [6]. It was shown that Galfenol exhibits higher magnetostriction with increasing Ga content up to ~ 20 at% Ga; however, Galfenol also exhibits the ductile-to-brittle transition at 15 at% Ga [7][8][9]. Therefore, ductile Galfenol wires with composition of 85 at% Fe and 15 at% Ga In cylindrical coordinates, the heater generates a radial temperature gradient, ∇ � �⃗ = ∇, throughout the Galfenol wire and a magnetic field in the axial direction, � �⃗ =, is applied.
Following Eq. 1 and 2, an azimuthal electric field is produced at every point along the coil, therefore a voltage drop, , across the wire is produced that scales with the wire length.
Glyptal enamel is used for the demonstration of large voltage generation due to its higher working temperature and thinner insulation, while PTFE is used for the estimation of the Nernst coefficient from simulations due to its better known physical parameters. The Nernst voltage is measured using a custom-made system including an electromagnet from SES Instruments and Keithley 2700.  Qualitatively identical behavior is observed in samples with different lengths.
Accurate measurement of the magnetization of the coil device is unfeasible using standard magnetometry methods due to the large sample size and coil geometry. Instead, we measure the local total field (Bl) for qualitative magnetization behavior by placing a digital teslameter in the − plane (perpendicular to �⃗ ) in close proximity to the coil end. The local applied field (Hl) is obtained with the coil removed. The local magnetization (Ml) is therefore determined by subtracting the local applied field from the local total field, µ0Ml=(Bl-µ0Hl), plotted in the top left inset of Fig. 1(b). The Ml-Hl behavior reveals a non-zero remanence and Ml does not saturate under the maximum applied field of 0.18 T. We ascribe this hard axis behavior to the shape anisotropy due to the coiled wire geometry. Although it is expected that the anomalous Nernst voltage dominates in ferromagnets before reaching saturation magnetization, the difference in linearity between the Nernst and magnetization data suggest contributions from both ordinary and anomalous Nernst effects. The zero-field corrected Nernst voltage, as a function of magnetic field and radial temperature gradient, is shown in Fig. 1(c). Line cuts along the y-axis at a fixed radial temperature gradient show a linear relationship between the Nernst voltage and magnetic field (blacked dashed line represents the data shown in Fig. 1(b) insulation, as shown in Fig. 3

(a). An increase in
Nernst voltage at any given magnetic field is observed by increasing the wire length. This is more clearly shown in Fig. 3(b), where the length-dependent Nernst voltage is plotted directly. The Nernst voltage is linear with L at different thermal gradients, demonstrating voltage scalability in the Nernst coil.
This wire geometry also serves as an alternative way to measure the Nernst coefficient for the materials that are already in the wire form. However, since polymer based materials are used to isolate the Galfenol wire electrically from the cartridge heater, most of the temperature drop is across the Glyptal or PTFE due to their low thermal conductivities. This makes a direct temperature measurement across the Galfenol wire both impractical, due to its geometry, and unreliable, due to the fraction of a degree temperature drop expected across it. Additionally, an external air flow directed towards our setup was introduced to serve as active cooling and aid temperature stabilization, making the heat flow azimuthally dependent and difficult to estimate. Therefore, finite-element method software COMSOL [11] is used to simulate the temperature profile in the Galfenol devices with PTFE insulation using the parameters from [12][13][14][15][16][17][18]. The details of the thermal modeling and input parameters are discussed in the supplementary materials.
The Nernst coefficient defined as ≡ This is compared to the results (red stars) from another independent measurement on a 6 × 2 × 1 mm bulk polycrystalline Galfenol with a Ga content of 18.4% obtained from the same vendor measured between ± 1.4 T in a custom-made cryostat. The measurement setup is similar to the one used in Ref. [19] and is also used for Galfenol thermal conductivity and specific heat measurements as discussed in the supplementary material.
The Nernst thermopower of the bulk sample, , at 82 K and 300 K shown in Fig. 4 inset does not fully saturate up to 1.4 T and is similar to the coiled wire measurement. The Nernst coefficient in the Galfenol wire (Ga15Fe85) is found to be slightly lower (by ~ 20%) than the one measured in the bulk sample (Ga18.4Fe81.6) at around room temperature possibly due to the difference in their magnetostriction. To the best of our knowledge, this is the first measurement on the Nernst coefficient of Galfenol. This value is higher than the reported N of 0.189 μV/KT in Ref. [20], 0.4 μV/KT in Ref. [5], and 0.56 μV/KT in Ref. [21] for single crystal Fe3O4 bulk, single crystal Fe bulk, and epitaxial FePt thin film, respectively, near room temperature. The Nernst coefficients in Ref. [20] and [21] are calculated based on the saturation magnetization ( 0 ) of the sample and are converted to saturation magnetic field ( 0 ) to compare to our result. The Nernst coefficient in the bulk Galfenol sample continuously increases in magnitude from 80 to 300 K. The physical origin of the temperature dependence requires further investigation.
In conclusion, a coiled wire wrapped around a cylindrical heater is experimentally Finite-element method modeling of the temperature profile in Galfenol with PTFE shrink tube Finite-element method software COMSOL [10] is used to simulate the temperature profile in the Galfenol wire with PTFE shrink tube. Fig. S1(a) Table S1.
These parameters are assumed to be temperature independent due to the relative small absolute temperature change in the measurement. The thermal properties of Galfenol are measured from a separate custom-made cryostat setup, as in Ref. [1], on a bulk sample from the same vendor with a very slight difference in composition. The density of Galfenol is estimated based on the atomic percent of Fe (85%) and Ga (15%). Fig. S1(b) shows a good agreement between measured and simulated temperature at the same point (gray pentagon in Fig. S1(a)) on the coil at different cartridge heating powers. Simulations in Fig. S1(c) show the average temperature (Taverage) and