Phonon hydrodynamics is a theoretical framework for predicting nondiffusive heat transport processes in solids at the nanoscale or under high-frequency excitations. This article presents the microscopic and thermodynamic foundations of the theory and reviews its applications. First, we discuss historical and modern derivations of hydrodynamic heat transport equations from the phonon Boltzmann transport equation (BTE), and highlight advanced methods to predict hydrodynamic effects from direct solutions of the BTE beyond the Relaxation Time Approximation. Then, we review the main experiments that uncovered nondiffusive heat transport effects and their interpretation from the hydrodynamic perspective. Overall, the developments summarized in this work establish phonon hydrodynamics as a vital tool for understanding and engineering thermal transport at the nanoscale in data-processing and energy-conversion devices.

Prominent phonon hydrodynamic phenomena were predicted in graphene at low temperatures due to the abundance of momentum-conserving three-phonon interactions. Recent studies, however, have shown that higher-order interactions constitute an additional resistive channel that significantly reduces the thermal conductivity of this material. Here, we show that the occurrence of hydrodynamic effects in graphene is severely conditioned by four-phonon interactions. Contrary to conventional understanding, we first demonstrate that the collective limit assumption, in which the phonon distribution is fully correlated, is not adequate to understand the hydrodynamic transport mechanisms in graphene. Then we report the key hydrodynamic parameters, namely the nonlocal length and the heat flux relaxation time, and we show that they are significantly reduced if considering full anharmonicity. Finally, we discuss observable implications in a variety of experimental configurations and we critically review previous predictions on the necessary conditions for the manifestation of collective phonon behavior and phonon hydrodynamics.

The microscopic constraints required for the emergence of hydrodynamic-like heat conduction in semiconductors as predicted by the Guyer-Krumhansl equation (GKE) are determined through energy-based deviational Monte Carlo simulations of the Boltzmann Transport Equation (MC-BTE). We simulate the process of heat release from a nanoscale heat source towards a semi-infinite Silicon substrate in steady-state by solely considering resistive phonon collisions with an average mean free path. We obtain good agreement between the microscopic (MC-BTE) and mesoscopic (GKE) approaches in capturing significant deviations from diffusive transport, such as the emergence of vorticity. The analysis shows that heat vortices appear in the vicinity of sufficiently small energy sources in the absence of momentum-conserving scattering. The GKE is able to capture the resulting non-diffusive transport effects at moderate Knudsen numbers in consistency with previous experimental work.

With wave-packet propagation simulations and heat flux estimation via molecular dynamics, we show that the heat flux radial distribution in silicon nanowires can be described by a mesoscopic model, the hydrodynamic heat equation. We observe Poiseuille like heat flux profile, that cannot be described by a simple kinetic model such as the Fuchs-Sondheimer model, in both pristine and core/shell nanowires. The addition of a shell does not change the shape of the radial heat flux distribution, but just modifies the maximum of the heat flux in the center of the nanowire. These results show that there is a heat flux depletion length for pristine or core shell nanowires, 1-2 nm away from the boundary of the crystalline part. The parameters of the mesoscopic model are discussed in terms of microscopic properties, including the phonon mean free path as function of frequency and the partial vibrational density of states in the different regions of the nanowire. (c) 2022 Elsevier Ltd. All rights reserved.

The temperature profile around nano- and microscale heat sources on silicon substrate is technologically important for many integrated circuit applications. Here we present full-field thermal imaging of heater lines and rings with different sizes at temperatures ranging from 100 to 300 K. We show significant deviations compared to Fourier's law which are both size and geometry dependent. This can be explained by a hydrodynamic model for heat transport in silicon based on the Guyer and Krumhansl equation using ab initio calculated parameters within the kinetic collective model framework. One of these parameters, the nonlocal length, is shown to quantitatively determine the situations where nonFourier behavior occurs. This length scale is some orders of magnitude smaller than the longest phonon mean free paths. Ballistic phonons are shown not to manifest directly in these experiments, thus indicating the failure of the multiscale relaxation time approximation. Furthermore, we discuss the differences between Hamiltonian/microscopic and entropic/mesoscopic strategies to address nanoscale heat transport, and the relations between phonon dynamics and thermodynamics. Finally, the nonequilibrium phonon distribution function is used to determine conditions under which hydrodynamic modeling can be used.

We apply a recently developed method for solving the linearized phonon Boltzmann equation to study the hydrodynamic thermal transport in dielectrics in the collective limit, i.e., when normal collisions dominate resistive ones. The method recovers Guyer and Krumhansl results for a single Debye branch and extends them to general dispersion relations and branches. Specifically, we obtain explicit microscopic expressions for the phonon distribution and for the transport coefficients in this limit. We find that the phonon distribution differs from the commonly used displaced distribution in two terms: one accounting for viscous flow and another one which allows us to solve a long-standing issue on drifting and driftless second-sound velocities. Thus, the new method allows us to generalize previous results and fill some gaps on fundamental aspects of the collective limit through a simple mathematical formalism. We compare the hydrodynamic framework with previous models and discuss its limitations.

We obtain some exact solutions in the context of solitons, for heat conduction with inertia along a cylinder whose heat exchange with the environment is a non-linear function of the difference of temperatures of the cylinder and the environment, due to a flux-limiter behavior of the exchange. We study the consequences of heat transfer and information transfer along the wire, and we compare the situation with analogous solitons found in nonlinear lateral radiative exchange studied in some previous papers. We also find further exact solutions in terms of Weierstrass elliptic functions for the sake of completeness.

Heat management is crucial in the design of nanoscale devices as the operating temperature determines their efficiency and lifetime. Past experimental and theoretical works exploring nanoscale heat transport in semiconductors addressed known deviations from Fourier’s law modeling by including effective parameters, such as a size-dependent thermal conductivity. However, recent experiments have qualitatively shown behavior that cannot be modeled in this way. Here, we combine advanced experiment and theory to show that the cooling of 1D- and 2D-confined nanoscale hot spots on silicon can be described using a general hydrodynamic heat transport model, contrary to previous understanding of heat flow in bulk silicon. We use a comprehensive set of extreme ultraviolet scatterometry measurements of nondiffusive transport from transiently heated nanolines and nanodots to validate and generalize our ab initio model, that does not need any geometry-dependent fitting parameters. This allows us to uncover the existence of two distinct time scales and heat transport mechanisms: an interface resistance regime that dominates on short time scales and a hydrodynamic-like phonon transport regime that dominates on longer time scales. Moreover, our model can predict the full thermomechanical response on nanometer length scales and picosecond time scales for arbitrary geometries, providing an advanced practical tool for thermal management of nanoscale technologies. Furthermore, we derive analytical expressions for the transport time scales, valid for a subset of geometries, supplying a route for optimizing heat dissipation.

Second sound is known as the thermal transport regime where heat is carried by temperature waves. Its experimental observation was previously restricted to a small number of materials, usually in rather narrow temperature windows. We show that it is possible to overcome these limitations by driving the system with a rapidly varying temperature field. High-frequency second sound is demonstrated in bulk natural Ge between 7 K and room temperature by studying the phase lag of the thermal response under a harmonic high-frequency external thermal excitation and addressing the relaxation time and the propagation velocity of the heat waves. These results provide a route to investigate the potential of wave-like heat transport in almost any material, opening opportunities to control heat through its oscillatory nature.

We present a formalism to solve the phonon Boltzmann transport equation (BTE) for finite Knudsen numbers that supplies a hydrodynamic heat transport equation similar to the Navier-Stokes equation for general semiconductors. This generalization of Fourier's law applies in general cases, from systems dominated by momentum-preserving normal collisions, as is well known, to kinetic materials dominated by resistive collisions, where it captures nonlocal effects. The key feature of our framework is that the macrostate is described in terms of the heat flux and its first derivatives. We obtain explicit expressions for the nonequilibrium phonon distribution and for the geometry-independent macroscopic parameters as a function of phonon properties that can be calculated from first principles. Ab initio model predictions are found to agree with a wide range of experiments in silicon. In contrast to approaches directly based on the BTE, the hydrodynamic equation can be solved in arbitrary geometries, thus providing a powerful tool for nanoscale heat modeling at a low computational cost.

Superlattices with scattering mechanisms at multiple length scales efficiently scatter phonons at all relevant wavelengths and provide a convenient route to reduce thermal transport. Here, we show, both experimentally and by atomistic simulations, that SiGe superlattices with well-established compositional gradients and a sufficient number of interfaces exhibit extremely low thermal conductivity. Our results reveal that the thermal conductivity of long-period (30-50 nm) superlattices with thicknesses below 200 nm is still thicknessdependent and higher than that of the corresponding alloy thin film. Increasing the number of periods up to 16 has a strong impact on heat propagation, leading to thermal conductivity values below the thin-film alloy limit. Lattice dynamics calculations confirm that the reduced thermal conductivity stems from the simultaneous effects of mass scattering, graded interface scattering, and coherent interference from the lattice periodicity. This study provides a significant step forward in understanding the role of compositional gradients in heat transport across nanostructures. The strategy of employing long-period graded superlattices with extremely low thermal conductivities has great potential for micro- and nano-thermoelectric generation and cooling of Si-based devices.

Non-uniform self-heating and temperature hotspots are major concerns compromising the performance and reliability of submicron electronic and optoelectronic devices. At deep submicron scales where effects such as contact-related artifacts and diffraction limits accurate measurements of temperature hotspots, non-contact thermal characterization can be extremely valuable. In this work, we use a Bayesian optimization framework with generalized Gaussian Markov random field (GGMRF) prior model to obtain accurate full-field temperature distribution of self-heated metal interconnects from their thermoreflectance thermal images (TRI) with spatial resolution 2.5 times below Rayleigh limit for 530nm illumination. Finite element simulations along with TRI experimental data were used to characterize the point spread function of the optical imaging system. In addition, unlike iterative reconstruction algorithms that use ad hoc regularization parameters in their prior models to obtain the best quality image, we used numerical experiments and finite element modeling to estimate the regularization parameter for solving a real experimental inverse problem.

We employ thermoreflectance thermal imaging to directly measure the steady-state two-dimensional (2D) temperature field generated by nanostructured heat sources deposited on silicon substrate with different geometrical configurations and characteristic sizes down to 400nm. The analysis of the results using Fourier's law not only breaks down as size scales down, but it also fails to capture the impact of the geometry of the heat source. The substrate effective Fourier thermal conductivities fitted to wire-shaped and circular-shaped structures with identical characteristic lengths are found to display up to 40% mismatch. Remarkably, a hydrodynamic heat transport model reproduces the observed temperature fields for all device sizes and shapes using just intrinsic Si parameters, i.e., a geometry and size-independent thermal conductivity and nonlocal length scale. The hydrodynamic model provides insight into the observed thermal response and of the contradictory Fourier predictions. We discuss the substantial Silicon hydrodynamic behavior at room temperature and contrast it to InGaAs, which shows less hydrodynamic effects due to dominant phonon-impurity scattering.

The aim of this paper is to consider soliton propagation of heat signals along a cylinder whose heat exchange with the environment is a non-linear function of the difference of temperatures of the cylinder and the environment and whose heat transfer along the system is described by the Maxwell-Cattaneo equation. To find the soliton solutions we use the auxiliary equation method. Our motivation is to obtain and compare the speed of propagation, the maximum rate of information transfer, and the energy necessary for the transfer of one bit of information for different solitons, by assuming that a localized soliton may carry a bit of information. It is shown that a given total power (energy/time) may be used either to send a few bits in a fast way, or many bits in a slower way. This may be controlled by choosing the initial condition imposed at one end of the wire. (C) 2020 Elsevier Ltd. All rights reserved.

According to Fourier’s law, a temperature difference across a material results in a linear temperature profile and a thermal conductance that decreases inversely proportional to the system length. These are the hallmarks of diffusive heat flow. Here, we report heat flow in ultrathin (25 nm) GaP nanowires in the absence of a temperature gradient within the wire and find that the heat conductance is independent of wire length. These observations deviate from Fourier’s law and are direct proof of ballistic heat flow, persisting for wire lengths up to at least 15 μm at room temperature. When doubling the wire diameter, a remarkably sudden transition to diffusive heat flow is observed. The ballistic heat flow in the ultrathin wires can be modeled within Landauer’s formalism by ballistic phonons with an extraordinarily long mean free path.

The diameter dependence of the thermal conductivity of nanowires is usually modeled using Matthiessen's rule, by putting the mean free path of phonons equal to the diameter d of the nanowire. This results in a thermal conductivity kappa that decreases with decreasing d, due to the increase in boundary scattering. Recent molecular dynamics studies of heat transport in thin silicon nanowires have shown a nonmonotonic diameter dependence of kappa, where a decrease with decreasing d is followed by an increase to a value of kappa exceeding the bulk thermal conductivity. This increase of kappa was explained by an increase of the importance of hydrodynamic transport effects in the thinner wires, where the normal scattering by phonon-phonon interaction increases, but the Umklapp scattering decreases [Y. Zhou, X. Zhang, and M. Hu, Nano Lett. 17, 1269 (2017)]. Here, we study heat transport in thin nanowires of the compound semiconductor gallium-phosphide in the wurtzite crystal structure, using molecular dynamics simulations. A similar nonmonotonic d dependence of kappa is found as in silicon nanowires, but with a minimum in kappa occurring at a much larger diameter of d approximate to 8 nm instead of 2-3 nm.