Thermoelectric (TE) cooling offers advantages such as compact size, high stability, and vibrationless operation, making it desirable for various applications, particularly in cryogenic environments where precise temperature control is essential for electronic and quantum materials. Traditional TE cooling methods, primarily based on the Peltier effect, become less efficient at lower temperatures, often overwhelmed by Joule heating. Recent interest has focused on high-purity single-crystal semimetals for cryogenic TE applications. The Peltier effect utilizes temperature gradients generated along the current flow direction in structures with differing Peltier coefficients. While Peltier cooling has been achieved down to liquid nitrogen temperatures, reports at liquid helium temperatures are limited. Microscopic devices with engineered energy gaps, like superconducting junctions or quantum dots, can enhance Peltier cooling at very low temperatures. This research explores a different magneto-electro-thermal mechanism – the Ettingshausen effect – previously unstudied in microscopic devices. The Ettingshausen effect generates a transverse temperature gradient perpendicular to both current flow and an applied magnetic field, offering a potentially powerful route to cryogenic cooling. In metals, this effect is weak due to the energy dependence of charge carrier drift velocity. However, in compensated semiconductors and semimetals with similar electron and hole densities and mobilities, a much stronger effect is anticipated due to electron-hole pair accumulation and recombination. This work investigates the Ettingshausen effect and cooling using van der Waals (vdW) semimetals, specifically WTe₂, known for its high charge carrier mobility and near-compensation of electron and hole densities. The unique properties of vdW materials allow for easy integration into layered structures, enabling novel device functionalities.

Extensive research on thermoelectric phenomena, particularly the Seebeck and Nernst effects (where voltages or currents are generated by temperature gradients), has been conducted in microscopic vdW devices. These studies typically measure voltages induced by global temperature gradients or local heating using focused laser beams. However, direct investigation of electro-thermal processes like the Peltier and Ettingshausen effects, requiring temperature gradient measurements in response to applied currents, is significantly more challenging at the microscopic scale, especially at cryogenic temperatures. Prior work has demonstrated large Nernst and Ettingshausen effects in bulk WTe₂ above 20 K, but the mesoscopic behavior at cryogenic temperatures remained unexplored. This research gap highlights the need for a high-resolution thermal imaging technique to probe the microscopic mechanisms of the Ettingshausen effect in WTe₂ at liquid helium temperatures.

This study employed cryogenic thermal imaging using a superconducting quantum interference device on a tip (SQUID-on-tip, SOT) with a diameter of 110 nm. The SOT, functioning as a nanothermometer, scanned above the WTe₂ sample surface (at a height of 80 nm) in a helium exchange gas atmosphere. A high-quality WTe₂ single crystal was exfoliated and patterned into rectangular chambers with varying widths. To avoid 1/f noise, an AC current at a specific frequency was applied to the chambers connected in series. The resulting AC temperature change was imaged by the scanning SOT. Two contributions to the temperature change were considered: Joule heating (quadratic in current) and the Ettingshausen effect (linear in current). These were separated by lock-in measurements at the fundamental and second harmonic frequencies of the applied current. Joule heating was uniformly distributed across most of the chamber, with hotspots at current constrictions. The Ettingshausen effect generated a large temperature gradient transverse to the current and magnetic field directions. To determine actual excess temperature, a unipolar square-wave current modulation was employed to measure the temperature difference between the current-on and current-off states. Three-dimensional finite-element numerical simulations, incorporating heat diffusion equations for WTe₂ and the Si substrate, were performed to model the experimental observations and provide insight into the underlying physical mechanisms. An analytical model was also developed to further understand the role of different parameters in the mesoscopic device.

The study demonstrated the Ettingshausen effect and achieved absolute cooling (temperature below the base temperature) in exfoliated WTe₂ flakes at 4.3 K. Joule heating, observed at the second harmonic, increased quadratically with current and showed a strong magnetic field dependence (quadratic with B), consistent with the large magnetoresistance of WTe₂. The Ettingshausen effect, detected at the fundamental frequency, generated a transverse temperature gradient that was linearly dependent on current but exhibited a non-monotonic dependence on both magnetic field and chamber width. The non-monotonic behavior in magnetic field and chamber width is attributed to the interplay of mesoscopic length scales and the non-uniform current distribution, which are absent in bulk measurements. Absolute cooling was observed at low currents and specific magnetic fields. The current threshold for absolute cooling decreased monotonically with increasing magnetic field, and the maximum cooling had a non-monotonic dependence on the magnetic field. Numerical simulations, using a three-dimensional model incorporating the heat transport equations, accurately replicated the experimental temperature profiles and the phase diagram for absolute cooling, providing values for key material parameters like electron-hole recombination length (around 0.5 μm) and mobility (around 25,000 cm²/Vs). An analytical model further confirmed the mesoscopic nature of the Ettingshausen effect, explaining the non-monotonic dependence of the temperature difference on chamber width and the sublinear dependence on magnetic field. This model introduced a new mesoscopic length scale (W₀) arising from the competition between in-plane and out-of-plane heat conductivities, a characteristic absent in bulk materials. This mesoscopic length scale and its relationship with device parameters provide insights for optimizing the cooling efficiency. The analytical model also successfully predicted the observed cooling regime and the optimal parameters for achieving the lowest temperatures.

The findings directly address the research question by demonstrating cryogenic magneto-thermoelectric cooling via the Ettingshausen effect in mesoscopic WTe₂ devices. The non-monotonic behavior of the cooling with respect to magnetic field and device size, a key finding, underscores the importance of considering mesoscopic effects in TE devices. The agreement between the experimental data and both the numerical and analytical models validates the proposed mechanism of electron-hole pair generation and recombination as the primary driver of the observed cooling. The identification of a new mesoscopic length scale (W₀) offers a crucial parameter for future optimization of TE cooling devices. The results have significant implications for the field of cryogenic cooling, suggesting a route towards integrating microscopic cooling elements directly into van der Waals devices for precise thermal management of electronic and quantum materials. The observed non-local behavior of the Ettingshausen effect further highlights the potential for controlling thermal landscapes at the nanoscale.

This study successfully demonstrates nanoscale cryogenic thermal imaging of the magneto-thermoelectric Ettingshausen effect and achieves absolute cooling in WTe₂ at liquid helium temperatures. The non-monotonic behavior observed is attributed to mesoscopic effects, successfully modeled by both 3D numerical and simplified analytical approaches. A novel mesoscopic length scale is identified, providing guidance for optimizing device geometry and material properties to enhance cooling efficiency. Future research should focus on improving material quality to reduce sheet resistance and further enhance cooling performance. Investigating other vdW semimetals and heterostructures could also broaden the applicability of this approach.

The study's limitations include the influence of surface oxidation on the WTe₂ flakes, reducing their mobility compared to bulk crystals. This effect leads to slightly different fitting parameters between experiment and simulation. Furthermore, the analytical model, while capturing the essence of the mesoscopic behavior, is a simplification of the full 3D system. Despite this, the results provide valuable insights into the fundamental mechanisms of magneto-thermoelectric cooling at the microscopic level.