Phonon Hydrodynamics Research Articles

Hydrodynamic heat conduction based on eigenvalues and eigenvectors of the normal collision operator

Abstract Although the Guyer-Krumhansl equations has opened up the study of phonon hydrodynamics in ultra-low temperature and low dimensional non-metallic crystals, it still cannot explain the high thermal conductivity of low dimensional non-metallic materials in adiabatic environments. In this work, the analytical solution of the linear Boltzmann transport equation with the Callaway approximation is obtained by expanding the nonequilibrium distribution function into a series of the orthogonal eigenvectors of the normal-process collision operator. By assuming the normal scatterings dominate the heat conduction in an anisotropic non-metallic crystal allowing the different branches of the phonon frequency spectrum having different group velocity, the macroscopic energy and momentum balance equations are developed for describing the phonon hydrodynamic transport. For an isotropic and dispersionless system, these balance equations reduce to the improved Guyer-Krumhansl equations. The thermal conductivity in these balance equations includes not only the contribution of the resistive scatterings, but also the contribution of the normal scatterings. Therefore, the improved Guyer-Krumhansl equations is capable for explaining the high thermal conductivity of suspended graphene, which is validated by the experimental results. Finally, the improved Guyer-Krumhansl equations is employed to derive the occurrence condition of the second sound in suspended single-layer graphene.

Non-Classical Heat Transfer and Recent Progress

Abstract Heat transfer in solids has traditionally been described by Fourier's law, which assumes local equilibrium and a diffusive transport regime. However, advancements in nanotechnology and the development of novel materials have revealed non-classical heat transfer phenomena that extend beyond this traditional framework. These phenomena, which can be broadly categorized into those governed by kinetic theory and those extending beyond it, include ballistic transport, phonon hydrodynamics, coherent phonon transport, Anderson localization, and glass-like heat transfer, among others. Recent theoretical and experimental studies have focused on characterizing these non-classical behaviors using methods such as the Boltzmann transport equation, molecular dynamics, and advanced spectroscopy techniques. In particular, the dual nature of phonons, exhibiting both particle-like and wave-like characteristics, is fundamental to understanding these phenomena. This review summarizes state-of-the-art findings in the field, highlighting the importance of integrating both particle and wave models to fully capture the complexities of heat transfer in modern materials. The emergence of new research areas, such as chiral and topological phonons, further underscores the potential for advancing phonon engineering. These developments open up exciting opportunities for designing materials with tailored thermal properties and new device mechanisms, potentially leading to applications in thermal management, energy technologies, and quantum science.

A graphite thermal Tesla valve driven by hydrodynamic phonon transport.

The Tesla valve benefits the rectification of fluid flow in microfluidic systems1-6 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 momentum-conserving collisions between phonons and the infrequent occurrence of liquid-like phonon flows. Recently, investigations and revelations of phonon hydrodynamics in graphitic materials7-10 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 micrometre-scale 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.

The Modified Guyer-Krumhansl Equations Derived From the Linear Boltzmann Transport Equation

Abstract Phonon hydrodynamics originated from the macroscopic energy and momentum balance equations (called Guyer-Krumhansl equations) proposed by Guyer and Krumhansl by solving the linearized Boltzmann transport equation for studying second sound in the normal-process collision dominated phonon transports in an isotropic nonmetallic crystal with a dispersionless frequency spectrum. However, the low-dimensional dielectric materials and semiconductors are anisotropic, and the different branches in their phonon frequency spectrum usually have different group velocities. For such materials, we derive the macroscopic energy and momentum balance equations from the linear Boltzmann transport equation to describe the phonon hydrodynamic transport, and solve the longstanding debate about whether the energy balance equation contains the second-order spatial derivatives of temperature. Finally, by solving the modified Guyer–Krumhansl equations, we find the minimum and maximum values of the length required by the occurrence of second sound in suspended single-layer graphene with the rectangular shape.

Phonon hydrodynamics in bulk insulators and semimetals

Phonon hydrodynamics in bulk insulators and semimetals

Quantitative assessment of thermal resistance in graphene field-effect transistors based on a phonon hydrodynamics and van der Waals coupling approach

In the progressive miniaturization of graphene field-effect transistors (GrFETs), pronounced thermal boundary resistance (TBR) and increasing ballistic-diffusive thermal resistance pose formidable challenges to thermal management. Quantitative analysis of thermal resistance is crucial for advancing efficient thermal designs in GrFETs using state-of-the-art technology. The inherently low-dimensional structure nature of graphene means that thermal conduction in GrFETs, governed by phonon hydrodynamic transport and van der Waals (vdW) interactions, cannot be accurately described by traditional Fourier's law-based models or the single-mode relaxation time approximation. This study introduces an innovative thermal simulation framework that addresses these challenges by incorporating phonon hydrodynamics equations and considering weak vdW coupling. Our findings indicate that thermal conduction in GrFETs is predominantly facilitated by flexural (ZA) phonons, and their thermal transport efficiency is significantly influenced by substrate perturbations. Specifically, the TBR in GrFETs is closely associated with the weak vdW coupling between ZA phonons and substrate atoms. Furthermore, this study employs dimensionless thermal resistance ratios and Knudsen numbers to explore the impact of silicon's geometric dimensions, heat source size, and ballistic phenomena on the substrate's thermal resistance. These insights provide substantial theoretical support for the development of efficient thermal management strategies in GrFETs.

Thermal properties of carbon-based materials

Phonons and related processes govern the thermal properties of materials. They can be excited externally using light, heat, mechanical energy, etc. This article aims to elucidate the methodology used in studying thermal properties, specifically employing temperature-dependent Raman spectroscopy. However, unique challenges arise when applying these techniques to low-dimensional materials. We summarize theoretical and experimental studies highlighting the significance of phonon hydrodynamics—a phenomenon typically observed only at low temperatures in bulk materials, but now found at room temperature in 2D materials like graphene. This discovery calls for caution when utilizing temperature-dependent Raman spectroscopy-based techniques, such as the optothermal Raman method, which has been extensively employed to experimentally determine the thermal conductivity of graphene. Moreover, temperature-dependent Raman spectroscopy remains a valuable tool for investigating phonon anharmonicity, a key factor governing phonon transport and decay processes in a wide array of emerging 2D materials and their heterostructures.

A power-law model for nonlinear phonon hydrodynamics

A power-law model for nonlinear phonon hydrodynamics

Ultrafast electron diffuse scattering as a tool for studying phonon transport: Phonon hydrodynamics and second sound oscillations.

Hydrodynamic phonon transport phenomena, like second sound, have been observed in liquid helium more than 50 years ago. More recently second sound has been observed in graphite at over 200 K using transient thermal grating (TG) techniques. In this work, we explore signatures of phonon hydrodynamic transport and second sound oscillations in ultrafast electron diffuse scattering patterns, which can provide time, momentum, and branch resolved information on the state-of-excitation of the phonon system beyond that available through TG experiments. We use the density functional theory and solve the Boltzmann transport equation to determine time-resolved non-equilibrium phonon populations and model phonon transport in graphite. This model also provides the information necessary to calculate the time evolution of one-phonon structure factors and diffuse scattering patterns during thermal transport covering ballistic, diffusive, and hydrodynamic regimes where the effect of a second sound oscillation on the phonon distribution is observed. Direct measurements of how the phonon distribution varies in time and space in various thermal transport regimes should yield new insights into the fundamental physics of the underlying processes.

Non-fourier phonon heat transport in graphene nanoribbon field effect transistors based on modified phonon hydrodynamic lattice Boltzmann method

Graphene, with its exceptional thermal conductivity and electron mobility, is widely recognized as a potential candidate for next-generation transistor materials. Despite the lack of pronounced phonon hydrodynamic phenomena at room temperature in graphene materials, momentum-conserving phonon normal scattering predominantly drives phonon hydrodynamic transport. Current methodologies, including the Single Mode Relaxation Time (SMRT) approximation and Fourier's law, inadequately capture phonon transport within the ballistic and hydrodynamic regimes of graphene. In light of these limitations, this study introduces a Non-gray phonon hydrodynamic lattice Boltzmann method in combination with the Temperature Jump Boundary Condition to provide a more accurate depiction of phonon heat transport in graphene nanoribbon field-effect transistors (GNRFETs) spanning these regimes. The findings reveal that our proposed model significantly mitigates heat transfer errors in GNRFETs, primarily driven by ballistic transport compared to the SMRT model. This suggests that the model's applicability extends beyond scenarios dominated by phonon hydrodynamics and is effective even in cases dominated by ballistic transport. In conclusion, our study emphasizes the impact of hydrodynamic transport on heat transfer in GNRFETs and underscores the importance of adjusting parameters related to the heating zone width and specular parameter to enhance thermal stability.

Second sound of heat conduction in one-dimensional dielectric materials

Although recent experiment have shown that second sound can occur in graphite above 200 K, there have been no reports of second sound being observed in low-dimensional materials. In the present work, based on phonon hydrodynamics we found that second sound can occur in a single-walled carbon nanotube (SWCNT) with a length of no less than 2.1333 microns and no more than 2.1209e-4 meters for the initial temperature field with sinusoidal changes in the axial direction. The constraint conditions for relaxation times of the normal and resistive scatterings, as well as the conditions for the axial length and initial temperature distribution required for the occurrence of the second sound in dielectric nanowires are also derived from the Guyer-Krumhansl equations. For both SWCNTs and nanowires it was found that the small normal scattering relaxation time and large resistive scattering relaxation time are beneficial for the occurrence of second sound. Our results show that in comparison with two-dimensional materials, such as graphene, it is easier to experimentally excite second sound in the SWCNTs.

Effect of atomic substitution and structure on thermal conductivity in monolayers H-MN and T-MN (M = B, Al, Ga).

Finding materials with suitable thermal conductivity (κ) is crucial for improving energy efficiency, reducing carbon emissions, and achieving sustainability. Atomic substitution and structural adjustments are commonly used methods. By comparing the κ of two different structures of two-dimensional (2D) IIIA-nitrides and their corresponding carbides, we explored whether atomic substitution has the same impact on κ in different structures. All eight materials exhibit normal temperature dependence, with κ decreasing as the temperature rises. Both structures are single atomic layers of 2D materials, forming M-N bonds, with the difference being that H-MN consists of hexagonal rings, while T-MN consists of tetragonal and octagonal rings. 2D IIIA-nitrides provide a good illustration of the impact of atomic substitution and structure on κ. On a logarithmic scale of κ, it approximates two parallel lines, indicating that different structures exhibit similar trends of κ reduction under the same conditions of atomic substitution. We analyzed the mechanisms behind the decreasing trend in κ from a phonon mode perspective. The main reason for the decrease in κ is that heavier atoms lower lattice vibrations, reducing phonon frequencies. Electronegativity increases, altering bonding characteristics and increasing anharmonicity. Reduced symmetry in complex structures decreases phonon group velocities and enhances phonon anharmonicity, leading to decreased phonon lifetimes. It's noteworthy that we found that atomic substitution and structure significantly affect hydrodynamic phonon transport as well. Both complex structures and atomic substitution simultaneously reduce the effects of hydrodynamic phonon transport. By comparing the impact of κ on two different structures of 2D IIIA-nitrides and their corresponding carbides, we have deepened our understanding of phonon transport in 2D materials. Heavier atomic substitution and more complex structures result in reduced κ and decreased hydrodynamic phonon transport effects. This research is likely to have a significant impact on the study of micro- and nanoscale heat transfer, including the design of materials with specific heat transfer properties for future applications.

Thermal rectification induced by phonon hydrodynamics in asymmetric 2D microstructures

A thermal diode/rectifier, where the thermal resistance is direction dependent, has always been highly sought after and may find applications in potential thermal logic devices and thermal management systems. By using a more accurate local spatial linear approximation (LSLA) when numerically solving the phonon Boltzmann transport equation (BTE) under Callaway's dual model, it is shown here for the first time that phonon hydrodynamics combined with geometric asymmetry can lead to thermal rectification (TR). Interestingly, in a T-shaped graphene microribbon heat preferentially flows from the narrow end to the wide end, which is opposite to what is observed in T-shaped graphene nanoribbons. Flexural acoustic (ZA) phonons play the dominant role in the TR effect, due to their prominent hydrodynamic behavior. By analysis of the temperature spatial distribution and the phonon mean free paths (MFP), the reasons for TR are explained. By adjusting geometric parameters, the 16 % TR ratio is reached. Furthermore, the impact of edge roughness, temperature difference and strength of phonon hydrodynamics on the TR effect are investigated. This work points to a new mechanism for enabling thermal rectification and may provide insight into the unique interplay between phonon hydrodynamics and engineered microstructures.

Thermal conductivity of group IV elemental semiconductors

The thermal conductivity of group IV elements—germanium, silicon, and diamond—is described in order to demonstrate various important and interesting aspects of the mechanism of phonon heat transfer in single-crystalline semiconductors and dielectrics. The measured temperature dependence of thermal conductivity κ(T) for these materials reveals different phonon scattering processes that determine thermal conductivity. In addition to the intrinsic processes of phonon–phonon scattering, scattering by isotopes, dopants, free electrons, sample surfaces, the effects of phonon focusing, irradiation with high-energy particles, and phonon hydrodynamics are discussed.

Ab initio investigations on hydrodynamic phonon transport: From diffusion to convection

In classical theory, heat conduction in solids is regarded as a diffusion process driven by a temperature gradient, whereas fluid transport is understood as convection process involving the bulk motion of the liquid or gas. In the framework of ab initio theory, which is directly built upon quantum mechanics without relying on measured parameters or phenomenological models, we observed and investigated the fluid-like convective transport of energy carriers in solid heat conduction. Thermal transport, carried by phonons, is simulated in graphite by solving the Boltzmann transport equation using a Monte Carlo algorithm. To capture convective transport, with phonon distributions deviating significantly from equilibrium Bose-Einstein distribution, we determined phonon interactions using ab initio approaches that go beyond relaxation time approximations. The presence of strong momentum-conserved Normal scatterings in graphite introduces a regime for hydrodynamic phonon transport. Fluid-like features, such as vortex and jet flow, are visualized and compared with classical theories on heat diffusion and fluid convection. Our study on phonon convection enhances fundamental understandings of heat conduction in solids from both atomic scale and quantum aspects, innovating thermal designs for future microelectronic devices and other thermal management applications. This potentially offers solutions for heat dissipation challenges in the post-Moore era.

Anomalous size effect of thermal conductivity of two-dimensional dielectric materials

In this work, we find that two-dimensional characteristics of hydrodynamic phonon transports in two-dimensional dielectric materials lead to the anomalous length dependency and even divergence of thermal conductivities. For suspended monolayer graphene the analytical solution of the two-dimensional Guyer-Krumhansl equations shows that the thermal conductivity increases with the increase of its length from 1 micron to 9 micron. When a specific temperature distribution at the inlet boundary is given, and the positive partial derivative of the heat flux along the length direction at the inlet and outlet boundaries is prescribed, analytical results show that the thermal conductivity of suspended monolayer graphene at the room temperature tends to infinity with the increase of its length. This result can be used to artificially regulate the thermal conductivity of suspended single-layer graphene.

Multiscale heat transport with inertia and thermal vortices

In this paper, we present a Hamiltonian and thermodynamic theory of heat transport on various levels of description. Transport of heat is formulated within kinetic theory of polarized phonons, kinetic theory of unpolarized phonons, hydrodynamics of polarized phonons, and hydrodynamics of unpolarized phonons. These various levels of description are linked by Poisson reductions, where no linearizations are made. Consequently, we obtain a new phonon hydrodynamics that contains convective terms dependent on vorticity of the heat flux, which are missing in the standard theories of phonon hydrodynamics. Within the zero-order Chapman-Enskog reduction, the resulting hydrodynamic equations are hyperbolic and Galilean invariant, while the first Chapman-Enskog expansion gives additional viscous-like terms. The vorticity-dependent terms violate the alignment of the heat flux with the temperature gradient even in the stationary state, which is expressed by a Fourier-Crocco equation. Those terms also cause that temperature plays in heat transport a similar role as pressure in aerodynamics, which is illustrated on numerical simulations of flow past a cylinder. In particular, we show that the vorticity-dependent terms lead to a colder spot just behind the cylinder, and for high-enough Reynolds numbers they lead to the von Kármán vortex street.

Non-Local Vectorial Internal Variables and Generalized Guyer-Krumhansl Evolution Equations for the Heat Flux.

In this paper, we ask ourselves how non-local effects affect the description of thermodynamic systems with internal variables. Usually, one assumes that the internal variables are local, but that their evolution equations are non-local, i.e., for instance, that their evolution equations contain non-local differential terms (gradients, Laplacians) or integral terms with memory kernels. In contrast to this typical situation, which has led to substantial progress in several fields, we ask ourselves whether in some cases it would be convenient to start from non-local internal variables with non-local evolution equations. We examine this point by considering three main lengths: the observation scale R defining the elementary volumes used in the description of the system, the mean free path l of the microscopic elements of the fluid (particles, phonons, photons, and molecules), and the overall characteristic size L of the global system. We illustrate these ideas by considering three-dimensional rigid heat conductors within the regime of phonon hydrodynamics in the presence of thermal vortices. In particular, we obtain a generalization of the Guyer-Krumhansl equation, which may be of interest for heat transport in nanosystems or in systems with small-scale inhomogeneities.

Phonon Hydrodynamic Transport: Observation of Thermal Wave-Like Flow and Second Sound Propagation in Graphene at 100 K.

Several experimental and theoretical investigations confirm the failure of the classical Fourier's law in low-dimensional systems and ultrafast thermal transport. Hydrodynamic heat transport has been recently considered as a promising avenue to thermal management and phonon engineering in graphitic materials. Non-Fourier features are therefore required to describe and distinguish the hydrodynamic regime from other heat transport regimes. In this work, we provide an efficient framework for the identification of hydrodynamic heat transport and second sound propagation in graphene at 80 and 100 K. We solve both the dual-phase-lag model and the Maxwell-Cattaneo-Vernotte equation based on the finite element method with ab initio data as inputs. We emphasize on the detection of thermal wave-like behavior using macroscopic quantities including the Knudsen number and second sound velocity beyond Fourier's law. We present a clear observation of the crossover phenomena from the wave-like regime to diffusive heat transport predicted in terms of mesoscopic equations. This present formalism will contribute to a clear and deeper understanding of hydrodynamic heat transport in condensed systems for future experimental detection of second sound propagation above 80 K.

Super-Ballistic Width Dependence of Thermal Conductivity in Graphite Nanoribbons and Microribbons.

The super-ballistic temperature dependence of thermal conductivity, facilitated by collective phonons, has been widely studied. It has been claimed to be unambiguous evidence for hydrodynamic phonon transport in solids. Alternatively, hydrodynamic thermal conduction is predicted to be as strongly dependent on the width of the structure as is fluid flow, while its direct demonstration remains an unexplored challenge. In this work, we experimentally measured thermal conductivity in several graphite ribbon structures with different widths, from 300 nm to 1.2 µm, and studied its width dependence in a wide temperature range of 10-300 K. We observed enhanced width dependence of the thermal conductivity in the hydrodynamic window of 75 K compared to that in the ballistic limit, which provides indispensable evidence for phonon hydrodynamic transport from the perspective of peculiar width dependence. This will help to find the missing piece to complete the puzzle of phonon hydrodynamics, and guide future attempts at efficient heat dissipation in advanced electronic devices.