Demonstration and imaging of cryogenic magneto-thermoelectric cooling in a van der Waals semimetal

Thermoelectric effects are valuable for power generation, cooling and thermometry due to their compactness, stability and tunability, but their efficiency degrades markedly at low temperatures where Joule heating dominates. Recent advances in high-purity single-crystal semimetals have renewed interest in cryogenic thermoelectricity. Conventional cryogenic cooling has focused on the Peltier effect, which requires structures with differing Peltier coefficients and has been demonstrated down to liquid nitrogen temperatures and only rarely near liquid helium temperatures. In contrast, the Ettingshausen effect—where a transverse temperature gradient arises from a longitudinal current in a perpendicular magnetic field—offers a different route. In compensated semimetals with nearly equal electron and hole densities and mobilities, the Hall field is suppressed and the Lorentz force drives both carrier types transversely, causing electron–hole accumulation/depletion near opposite edges and enhancing recombination/generation, respectively, producing heating and cooling localized within an electron–hole recombination length from the edges. Van der Waals semimetals are promising for mesoscopic integrated thermoelectric cooling due to ease of integration into heterostructures. WTe2, a Weyl semimetal with very high mobility and near charge compensation, exhibits large Nernst and Ettingshausen responses at elevated temperatures. This study asks whether Ettingshausen cooling can be realized and imaged in microscopic vdW devices at liquid helium temperatures, and how mesoscopic geometry, recombination length and thermal coupling govern the efficiency.

Prior work has demonstrated Peltier-based cooling in bulk materials down to liquid nitrogen temperatures, with rare reports at liquid helium temperatures, often relying on engineered energy filters (e.g., superconducting junctions or quantum dots) to enhance low-temperature efficiency. Semimetals and topological materials (e.g., NbSb, NbP, Mg3Bi2, Mg3Pb, ZrTe5) exhibit large thermoelectric and magneto-thermoelectric coefficients at cryogenic to intermediate temperatures, including giant Nernst responses. In bulk WTe2, ultrahigh Ettingshausen signals above ~20 K have been reported. Microscopic vdW devices have been extensively probed for thermoelectric effects that generate voltages from temperature gradients (Seebeck, Nernst), often using global temperature gradients or local optical heating. However, direct imaging of temperature gradients arising from applied current (Peltier/Ettingshausen) in microscopic devices has been limited to temperatures above liquid nitrogen. The mesoscopic theory of two-component conductors predicts edge-dominated current flow and magnetoresistance suppression near edges due to boundary conditions and finite recombination length, which should strongly affect Ettingshausen phenomena in confined geometries.

- Materials and device: High-quality bulk WTe2 crystals grown by flux method (residual resistance ratio ~3,250; magnetoresistance ratio ~62,000 at 9 T and 2 K; mean free path ~20 μm). Exfoliated flakes placed on Si/SiO2 (290 nm oxide), patterned by e-beam lithography and reactive ion etching into rectangular chambers of various widths (e.g., W = 5, 10, 20 μm) connected by constrictions; flake thickness example d = 269 nm. Contacts formed by Ti/Au after Ar ion milling to remove native oxide. Flake transport mobilities ~10,000–30,000 cm² V⁻¹ s⁻¹ at 4 K. - Cryogenic thermal imaging: A MoRe SQUID-on-tip (SOT) nanothermometer (diameter ~110 nm) scanned at height ~80 nm in He exchange gas at T0 = 4.3 K; field up to 5 T. Thermal sensitivity ~2.6 μK Hz⁻1/2 at zero field with weak field dependence. Imaging performed with 40 ms per pixel, 344×269 pixels (~90 nm pixel size). - Excitations and lock-in detection: To separate linear Ettingshausen and quadratic Joule contributions, a sinusoidal a.c. current (f = 85.37 Hz) is applied. Ettingshausen-induced temperature modulation (linear in I) detected at fundamental frequency; Joule heating modulation (quadratic in I) detected at second harmonic; d.c. Joule component not captured in harmonic imaging. To map total excess temperature relative to T0 (including d.c. Joule), a unipolar square-wave current was applied and the current-on/off temperature difference δT(x,y) was recorded; negative δT indicates absolute cooling below T0. - Data collected: Spatial maps of δT_J (Joule) and δT_E (Ettingshausen) versus B (0–5 T) and I (up to ~50 μA), across different chamber widths. Absolute cooling maps δT using square-wave excitation were acquired to identify parameter regimes with δT_cold < 0. - Numerical simulations: 3D finite-element (COMSOL) simulations of two-component transport (electrons and holes) in the device geometry to compute charge current J, quasiparticle current P, out-of-equilibrium quasiparticle density δn_q, and power densities Q_J = E·J and Q_E from generation/recombination and P work. Heat diffusion in WTe2, He gas, SiO2 and Si substrate solved to obtain temperature fields. Fitted microscopic parameters include mobility μ ≈ 25,000 cm² V⁻¹ s⁻¹, recombination length l ≈ 0.5 μm, and average phonon energy per e–h recombination ε_p ≈ 0.06 meV; thermal conductivities (e.g., K_WTe2 ≈ 6.4 W m⁻¹ K⁻¹; K_SiO2 ≈ 0.2 W m⁻¹ K⁻¹; K_Si ≈ 0.32 W m⁻¹ K⁻¹) used. - Analytical model: A simplified mesoscopic heat-flux framework relates the threshold current for absolute cooling I0(B) and the transverse Ettingshausen temperature difference ΔT_E to material/electrothermal parameters, incorporating in-plane and out-of-plane heat conductance, device width W, thickness d, substrate effective thickness d_s and conductivities K, K'. Key relations derived include I0 ∝ (1+μ²B²)(1+W²/W0²)/(μB) and a sublinear ΔT_E(B) with non-monotonic W dependence via a mesoscopic length W0 ≈ √8 d² K/K'.

- Direct nanoscale imaging of the Ettingshausen effect at 4.3 K in exfoliated WTe2 vdW devices, with spatially resolved transverse heating and cooling localized near edges and enhanced near constrictions. - Ettingshausen magnitude: For B = 5 T and W = 20 μm, an ultrahigh transverse signal ΔT_E/(I J W) ≈ 2.2 × 10⁻⁵ K A⁻¹ m, comparable to bulk reports but achieved at much lower temperature. - Nonlinear field and non-monotonic width dependence: ΔT_E is not linear in B; chambers with W = 10 μm exhibited larger ΔT_E than W = 20 μm or 5 μm, evidencing mesoscopic effects beyond bulk scaling (B/W). - Absolute cooling below base temperature: Using a unipolar square-wave drive, δT(x,y) maps show regions with δT < 0 at the cold edge and corners, demonstrating absolute Ettingshausen cooling at liquid He temperature. Cooling at the cold edge δT_cold(I,B) is negative over a finite parameter space; δT_cold exhibits non-monotonic dependence on both I and B due to competition between linear Ettingshausen cooling and quadratic Joule heating. Maximal cooling occurs near B ≈ 2 T in the W = 20 μm device. - Edge-dominated transport and dissipation: Simulations show magnetoresistance suppression at edges, leading to edge-peaked current channels of width l_R = l/√(1+μ²B²). Quasiparticle current P flows mainly transverse to J; δn_q sharply accumulates at boundaries, producing antisymmetric Ettingshausen power Q_E (cool at one set of edges, hot at the opposite), while Joule power Q_J is largest in constrictions and edge slivers. - Phase diagram and model validation: The experimental δT_cold(I,B) phase map agrees with both 3D simulations and an analytical model. The threshold current I0(B) for absolute cooling follows I0 = (ε_p ε_0/4μB)(1+μ²B²)(1+W²/W0²), with maximum I0/B at μB ≈ 1. - Extracted parameters: μ ≈ 25,000 cm² V⁻¹ s⁻¹ (from I0(B) and simulations; alternative device gave μ ≈ 9,000 cm² V⁻¹ s⁻¹), electron–hole recombination length l ≈ 0.3–0.5 μm, average phonon energy per recombination ε_p ≈ 0.06 meV, mesoscopic heat-conductance length W0 ≈ 24 μm. Sheet resistance ρ0 ≈ 0.54 Ω (effective bulk ρ0 d ≈ 1.5 × 10⁻⁵ Ω·cm for d = 269 nm), much higher than best bulk (ρ ≈ 2.3 × 10⁻⁷ Ω·cm), suggesting room for improvement. - Predicted maximum cooling: Analytical estimate δT_cold^max ≈ −0.2 mK for W = 20 μm near optimal μB, consistent with measurements. - Nonlocal character: Ettingshausen heating/cooling observed in regions with little current flow (e.g., circular chambers connected to narrow strips), underscoring nonlocal dependence on quasiparticle flow rather than local J.

The study demonstrates that in mesoscopic compensated semimetals, the Ettingshausen effect at cryogenic temperatures is governed by boundary conditions and finite electron–hole recombination length, which reshape current and quasiparticle flows. This leads to edge-localized heating/cooling and nontrivial dependence on device width and magnetic field, deviating from bulk ΔT_E ∝ B/W scaling. The direct imaging verifies the mechanism of transverse electron–hole accumulation/depletion and the resultant phonon absorption/emission near edges. Absolute cooling below 4.3 K is achieved within a specific current–field window where linear Ettingshausen cooling overcomes quadratic Joule heating; the accessible cooling regime and optimal parameters (μB ~ 1, W ~ W0) are captured by both simulations and a compact analytical model. These results validate the feasibility of integrating local, electrically controlled cryogenic cooling elements into vdW heterostructures, enabling on-chip temperature landscape engineering and potentially improved performance for sensitive quantum materials and devices.

This work introduces nanoscale cryogenic imaging of magneto-electro-thermal phenomena and provides the first demonstration of Ettingshausen cooling at liquid helium temperatures in a vdW semimetal device. The experiments and models reveal that cooling efficiency and temperature profiles in mesoscopic devices are controlled by geometry, recombination length, mobility, magnetic field, and thermal coupling to the substrate. A compact analytical framework and 3D simulations accurately describe the observed nonlinearity in B, non-monotonic width dependence, and the parameter space for absolute cooling. The extracted material parameters (μ, l, ε_p, W0) enable rational design. Future directions include reducing sheet resistance via inert-environment fabrication and hBN encapsulation, optimizing thermal pathways and device geometries, extending to other high-mobility compensated semimetals, and integrating atomic-layer thermoelectric coolers into complex vdW stacks for local temperature control and exploration of correlated states.

- The magnitude of absolute cooling is small (on the order of tenths of a millikelvin) and achievable only within limited current and field ranges, constrained by Joule heating. - Device resistivity is elevated relative to high-quality bulk due to surface oxidation and scattering in thin flakes, reducing cooling efficiency; the authors expect improvements with encapsulation and inert fabrication. - Numerical simulations, while in good qualitative agreement, rely on parameters that are not uniquely determined; several thermal and transport parameters admit a range of values, so extracted numbers carry uncertainty. - The study focuses on WTe2 devices and specific geometries; generality across materials and broader device architectures remains to be established. - Field range is limited to 5 T, and thermal calibration uncertainties may affect absolute temperature values in some datasets (notably in additional geometries).