The Tesla valve benefits the rectification of fluid flow in microfluidic systems 1-6 and inspires researchers to design modern solid-state electronic and thermal rectifiers referring to fluid-rectification mechanisms in a liquid-state context. In contrast to the rectification of fluids in microfluidic channels, the rectification of thermal phonons in micro-solid channels presents increased complexity owing to the lack of momentumconserving collisions between phonons and the infrequent occurrence of liquid-like phonon flows. Recently, investigations and revelations of phonon hydrodynamics in graphitic materials 7-10 have opened up new avenues for achieving thermal rectification.

Here we demonstrate a phonon hydrodynamics approach to realize the rectification of heat conduction in isotopically enriched graphite crystals. We design a micrometrescale Tesla valve within 90-nm-thick graphite and experimentally observe a discernible 15.2% difference in thermal conductivity between opposite directions at 45 K. This work marks an important step towards using collective phonon behaviour for thermal management in microscale and nanoscale electronic devices, paving the way for thermal rectification in solids.

In microfluidics, the Tesla valve, invented and patented by Nikola Tesla in 1920 (ref. 1), is widely used to manipulate fluid flow 2-6 . The inherent asymmetry of the Tesla valve leads to the distinct fluid-flow behaviours in opposite directions. In the reverse flow direction, the viscous fluid encounters notable impedance, whereas in the forward direction, the flow experiences considerably less resistance. Thus, it realizes the rectification of fluid flows without the need for any moving parts 2,6 .

Analogous to macroscopic phenomena in fluid dynamics, the collective motion of electrons or phonons manifests extraordinary hydrodynamic electrical 11-13 or thermal 14-16 conduction in solids. In particular, phonon hydrodynamics shows peculiar characteristics of collective behaviours, such as the second sound 8,9,17 , phonon Poiseuille flow 10,18,19 , phonon vortex 20-22 and phonon Knudsen minimum 23-26 . Despite the recent observations of hydrodynamic thermal phenomena, contemporary phonon hydrodynamics remains in an exploratory phase, with practical applications yet to be realized. This is primarily attributed to stringent observation conditions 7,27 and the limited availability of materials capable of hydrodynamic phonon transport 28,29 .

On the other hand, the rectification of thermal conduction has promising application in engineering thermal energy, demonstrating potential capability in the design of thermal diodes 30-32 . The thermal-rectification effect has been claimed in many solid systems comprising two components with different temperature-dependent thermal properties 33-35 . It allows stronger heat flux in one direction and weaker heat flux in the opposite direction. However, induction of external treatments is required to trigger a phase transition of the material 33 or atomic-scale synthesis is required to avoid further effects arising from heat conduction across interfaces between heterogeneous compounds 34 . Furthermore, trapezoidal and asymmetric nanoporous structures have been devised to achieve rectification of thermal conduction in homogeneous material systems 36-38 . However, either a relatively large temperature difference is required to achieve the thermal rectification 36,37 or the directivity of the thermal behaviour is limited to the demonstration of local rectification effects within nanometre/ submicrometre scales 36,38 .

In this study, we designed asymmetric structures inspired by the Tesla valve used in hydrodynamics (Fig. 1a). We fabricated micrometre-scale samples using graphite, a material known for its inherently robust hydrodynamic phonon behaviour 8,10,24 . Using the thermoreflectance technique, we examined the thermal properties and rectification effects exhibited by the suspended graphite Tesla valve across the temperature range of 10-300 K. Analogous to hydrodynamics, the thermal rectification observed in solid graphite is attributed to hydrodynamic-like phonon flows in the Tesla valve. The concept of a hydrodynamic phonon Tesla valve in this work holds promise for achieving thermal rectification within complex micrometre-scale structures, thereby introducing new prospects for thermal manipulation using phonon hydrodynamics.

In graphite, the out-of-plane phonon modes induced by the interlayer interactions of graphene are largely populated at the Γ point 39,40 . This leads to intense anharmonic collisions among phonons with small wavevectors, consequently promoting frequent normal scattering in a momentum-conserving manner 28,41 . The strength of such conservation of phonon momentum plays a crucial role in the maintenance