Nernst coefficient of Pt by non-local electrical measurement

In this work, we investigate the Nernst effect of a thin platinum strip from 3 to 300 K in this non-local geometry. In this case, the heat is generated via Joule heating by the injection of a DC charge current into the injector strip which is then injected into the substrate (Fig. 1). The heat then diffuses across the strip detector electrode separated from the injector electrode by a distance d and creates a thermal gradient across the width of the detector electrode. In the presence of a magnetic field perpendicular to the temperature gradient, a transverse electric field emerges in the detector electrode. YSZ (Y₂O₃ stabilized ZrO₂, 8 mol % Y₂O₃) and MgO substrates are used to tune the thermal gradient and the resulting Nernst signal. Together with finite element analysis for modeling heat flow and temperature dependent thermometry measurements, we determine the Nernst coefficient of Pt from room temperature (8.56  $nV K − 1 T − 1$⁠) to 10 K (29.3  $nV K − 1 T − 1$⁠).

First, magnon transport has been demonstrated in magnetic systems^1,2,9–12 Magnons can be excited by injecting spin with a heavy metal injector (such as Pt or W) via the spin Hall effect, and the excited magnons can then propagate to the detector where the magnon spin is converted back to a charge current by the inverse spin Hall effect. This is solely a first-order effect with no thermal contribution and can be excluded if the observed signal is only a second-order signal.

Next, the spin Seebeck effect arises from a temperature gradient across the separation distance between injector and detector electrodes, inside the substrate or film, driving a spin current (or magnon) toward the detector. Excited thermally, this is a second-order effect as the heating power is proportional to $I inj 2$⁠. It has been experimentally observed in ferromagnet and antiferromagnet materials using the non-local geometry with an applied in-plane magnetic field.^3,13–17 In a ferromagnet, the spin Seebeck signal would be nonlinear with magnetic field and eventually saturate as the field strength surpasses the coercive field. In an antiferromagnet, the spin Seebeck would show nonlinearity and saturation with field but typically at much higher field strengths. Furthermore, recent investigation of spinon transport reported by means of the spin Seebeck effect in the proposed quantum spin liquid system seem promising,¹⁸ yet, in this particular system, the reported signature of the spin Seebeck effect has a linear dependence on field that closely resembles the Nernst effect. Hence, a careful examination is needed such as control measurements to help us clarify the results.

Finally, the Nernst effect would occur in the non-local geometry from the temperature gradient across the width of the detector strip, contrary to the spin Seebeck effect where the signal is generated from the temperature gradient across the region between the injector and detector electrodes. One distinctive feature that can be used to discern the two is the orientation of the magnetic field. The Nernst effect requires the magnetic field to be orthogonal to the temperature gradient, and therefore, oriented out-of-plane.

We first confirm the properties of our Pt electrodes with temperature and field-dependent resistance measurements of the injector strip [Fig. 2(a)]. Conventional metallic behavior, negligible magnetoresistance (field is normal to the sample surface), and linear I(V) (inset) curves are observed. Contrary to this, Fig. 2(b) shows the raw detector voltage, V_det, where a clear parabolic dependence on the injection current and monotonic magnetic field dependence is observed. The V_asym signal [Fig. 2(c)] further highlights the thermal contribution where the second-order fit is in good agreement. The magnetic field dependence of V_asym matches well with the Nernst effect as the signal is proportional to the magnetic field and vanishes at zero field. Furthermore, the symmetry of the field dependence matches that of the Nernst effect. Figure 2(d) shows the rotational dependence of V_asym with different magnetic field strengths. A $cos(φ)$ dependence is observed, with the signal showing a maximum (minimum) when the magnetic field is applied out-of-plane. A constant background signal (thermovoltage) has been removed.

To further evaluate this magnetothermal effect, the R₂ values were extracted from the injection current sweeps at various out-of-plane field strengths [Figs. 3(a) and 3(b)]. R₂ shows a linear dependence in agreement with the Nernst effect and with no sign of saturation or non-linearity that is normally observed in magnon and spin Seebeck transport in magnetic media. YSZ has a low thermal conductivity (2.4 W m⁻¹ K⁻¹),¹⁹ to investigate the role of the thermal properties of the underlying substrate on the Nernst signal; we also perform comparative measurements on a (001)-oriented magnesium oxide (MgO) single crystal substrate [thermal conductivity = 45 W m⁻¹K⁻¹ (Ref. 20)] that has the same dimensions as the YSZ crystal. Consistent with the higher thermal conductivity, and thus, smaller thermal gradient across the detector electrode, Fig. 3 reveals a small (negligible) signal observed for the MgO sample. Finally, the measurement was conducted with a wide range of temperatures, from 3 to 300 K. The R₂ magnitude decreases with decreasing temperature approximately linearly until ∼20 K.

To understand this temperature behavior and quantify the Nernst coefficient, the changes in thermal properties of the substrate need to be taken into account. This is studied by tracking the temperature of the detector while heating up the injector. The detector temperature was evaluated from the temperature dependent resistance of the platinum wire. Figure 4(a) shows a clear temperature rise of the detector upon injecting current, with higher temperatures observed for the smaller device gaps, verifying the effectiveness of this thermometry probe.

For quantifying the Nernst coefficient, the thermal gradient across the platinum strip is a prerequisite. We compute this with finite element analysis. Our analysis makes use of the stationary study with the AC/DC module for modeling the Joule heating and heat transfer module for analyzing the temperature field, and our device geometry with the material's properties is listed in Methods. The fidelity of the model can be seen in Fig. 4(a), showing a good match between the measured detector temperature and the simulated value with less than 0.1% difference. Next, the temperature difference $ΔT$ across the width of the detector is computed as a function of spacing distance, for both the YSZ and MgO substrates shown in Fig. 4(b). The results confirm a much larger thermal gradient generated when the YSZ substrate is used compared with the MgO substrate, in agreement with the scaling of measured R₂ values for the samples on the two substrates. Note that the scaling of R₂ for the MgO sample is smaller than expected for the given temperature gradients of the YSZ and MgO samples. We attribute the lower R₂ values for the MgO relative to simulations to an appreciable interface thermal resistance, which can be significant at interfaces between thin metals and oxides.^21,22 Furthermore, MgO is well known to form hydroxylates on its surface when exposed to water, complicating the atomic scale structure of the interface.²³ The simulations shown in Figs. 4(a) and 4(b) are done without interface resistance and agree well with the data for the YSZ sample. The agreement is not as strong for the MgO sample as shown in the supplementary material, Fig. S1(a). Much better agreement is achieved when adding interfacial thermal resistance of 2 × 10⁻⁸K m²W⁻¹ (comparable in size to many other oxide metal interfaces²⁴). The additional resistance lowers the detector temperature, and more importantly, the temperature profile of the detector [supplementary material, Fig. S1(b)]. The temperature difference across the detector goes from 0.009 K without interfacial thermal resistance to ∼0.002 69 K with interfacial thermal resistance, revealing a more than 3× reduction in the temperature gradient in agreement with the R₂ data in Fig. 3(b).

In addition, the R₂ signal is tracked at multiple spacing distances, from 200 to 800 nm shown in Fig. 4(b). The difference in spacing leads to a different thermal gradient across the detector width, which agrees with the trend from simulation.

The thermometry measurement was conducted as a function of temperature to estimate the change in the temperature gradient. Figure 4(c) shows the measured temperature difference of the detector between the heated state (⁠ $I inj$ = 200 μA) and non-heated state (⁠ $I inj$ = 0 μA), exhibiting a similar temperature dependence to the thermal conductivity, implying that the temperature gradient is strongly affected by thermal conductivity. With this, $∂ xT$ was estimated using the simulated value at 300 K, then calibrated according to the thermometry probe [Fig. 4(c)]. Finally, putting together the thermal gradient obtained from the simulation work and the electrical measurements leads us to the Nernst coefficient of platinum, shown in Fig. 4(d). This reveals a small temperature dependent region above 30 K (with value of 3.14  $nV K − 1 T − 1$ at 30 K) and a sudden increase below that temperature. The high temperature regime (40–300 K) fits well to a linear function with a slope of 24.79 ± 0.989  $pV K − 1 T − 2$ and y-intercept of 1.491 ± 0.176  $nV K − 1 T − 1$⁠. The Nernst coefficient can also be obtained using different spacing d shown in the supplementary material, Fig. S2, where a variation of less than 10% is observed, suggesting an accurate evaluation.

Finally, we corroborate our experimentally determined value with an estimate from bulk material parameters for Pt and compare to values for metals in the literature. The Nernst coefficient varies greatly with the type of materials, with values in the range of 0–10 μV K⁻¹T⁻¹ for many metals,²⁵ to extremely large values reported for semimetal bismuth (above 1 mV K⁻¹T⁻¹).^26,27 Other reported values include 180  $nV K − 1 T − 1$ (20 K) for gold and 900  $nV K − 1 T − 1$ (20 K) for copper.²⁵ Yet, the Nernst coefficient for key materials, specifically, materials with strong spin–orbit coupling that are commonly used in spin transport measurements (platinum and tungsten), is missing.

In this work, we determine a clear magnetothermal signal in a non-local geometry, with two platinum strips with a thickness of 15 nm patterned on YSZ and MgO substrates. We confirmed the signal is attributed to the Nernst effect based on a systematic analysis, with the field and angular dependence matching the signatures of the Nernst effect. Together with the simulation work, we determined the Nernst coefficient for platinum at room temperature and below. Our work provides guidance for identifying various effects that could occur in such a configuration. In addition, we demonstrated a convenient approach for studying the Nernst effect in thin film where this work can be further extended and applied for heat transport.

See the supplementary material for methods and extended data.

This work was performed in part at the University of Michigan Lurie Nanofabrication Facility. The authors acknowledge the University of Michigan College of Engineering for financial support and the Michigan Center for Materials Characterization for use of the instruments and staff assistance.

The authors have no conflicts to disclose.

Tony Chiang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Johanna Nordlander: Conceptualization (equal); Writing – review & editing (supporting). Julia A. Mundy: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Writing – review & editing (supporting). John T. Heron: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.