Thermal transport in two-dimensional (2D) materials has attracted great attention since the discovery of high thermal conductivity in graphene, which is closely related to the hydrodynamic phonon transport. In this Perspective, we briefly summarize the recent progresses in studying hydrodynamic phonon transport in 2D materials, including both theoretical and experimental works. First, the criterion and numerical methods for studying hydrodynamic phonon transport are reviewed. We then discuss the physical mechanism and peculiar phenomena related to hydrodynamic phonon transport in 2D materials and finally present the challenge for future studies. This Perspective aims to provide the physical understanding of the hydrodynamic phonon transport, which might be beneficial to the exploration of novel thermal transport behaviors in 2D materials.

Thermal transport in two-dimensional (2D) materials is receiving increasing attention due to the extraordinary thermal properties and rich physics.1,2 Ultrahigh thermal conductivity has been recorded in 2D materials. For instance, the thermal conductivity (κ) of monolayer graphene is around 3000 W m−1 K−1 at room temperature, comparable to that of diamond.3 Besides, the χ 3 borophene has a high κ value around 512 W m−1 K−1 at room temperature along the armchair direction,4 similar to that of a typical metal. Interestingly, other 2D materials exhibit low κ values, such as phosphorene on the order of 10 W m−1 K−1 at room temperature in the armchair direction.5 The wide spreading κ values of the 2D materials can have promising applications in heat dissipation6 and thermoelectrics.7 

Unlike the size-independent κ of bulk materials in which diffusive thermal transport usually takes place, the size-dependent thermal transport is typically observed in 2D materials.8–10 In general, the well-known ballistic/diffusive thermal transport behavior emerges when the sample size is much shorter/longer than the phonon mean free path (MFP). In 2D materials, the phonon MFP can be affected by various factors, such as doping,11 defect,12 edge chirality,13 stress,14 and substrate coupling.15 Moreover, some interesting phonon transport phenomena, such as phonon interference,16–18 phonon localization,19,20 and thermal rectification,21–24 have also been reported in 2D systems. For example, Chen et al.18 found in numerical simulations that thermal conductivity of a Lorentz gas model oscillates with the degree of boundary roughness due to the interference between the transporting particles and the periodic boundary conditions. In contrast to the reduced thermal conductivity by isotopic doping, Ma et al.20 demonstrated via molecular dynamics simulations that thermal conductivity of the pillared graphene nanoribbon can be unexpectedly enhanced by the isotopic resonance at the pillars, which essentially breaks the phonon hybridization. Moreover, based on a two-dimensional multi-particle Lorentz gas model, Wang et al.24 observed in numerical simulations that the thermal rectification ratio has a scaling relation with the geometric parameters and source temperatures. Therefore, the diversity of 2D materials provides an ideal testbed to explore the various phonon transport behaviors.

Another unique phonon transport behavior, hydrodynamic phonon transport, has also been discovered in 2D materials, in which phonons propagate hydrodynamically in the form of collective drift motion.25,26 Hydrodynamic phonon transport provides a new perspective to understand thermal transport in solids beyond conventional diffusive and ballistic transport pictures. When the momentum-conserving normal (N) process is much stronger than the Umklapp (U) process, hydrodynamic phonon transport appears that can lead to a high κ. For instance, monolayer graphene, which has a quadratic phonon dispersion and prevailing N process for phonon scattering,27–30 shows an unusually high κ due to the hydrodynamic phonon transport. Inspired by the distinctive phonon dispersion, there have been increasing efforts recently to explore the phonon hydrodynamics in 2D materials.25,31–34

In this Perspective, we aim to provide a mini-review focusing on the recent advances on the understanding of hydrodynamic phonon transport in 2D materials. The rest of this paper is organized as follows. Section II introduces the criterion and numerical methods for hydrodynamic phonon transport. In Sec. III, we discuss the recent advances on phonon hydrodynamics in 2D materials. Finally, the conclusion and perspective on the challenges and opportunities are presented in Sec. IV.

In order to understand the regime in which phonon hydrodynamics occurs, one should inspect the typical physical mechanisms of internal scatterings: the anharmonic phonon scattering and isotopic scattering. The anharmonic phonon scattering includes the N and U processes. For the N process, the phonon momentum and total energy of the system are both conserved after the phonon collision. For the U process, the net momentum changes but the total energy remains conserved after the scattering. As a result, the U process contributes directly to the thermal resistance. In contrast, the N process does not contribute directly to thermal resistance but can indirectly affect the thermal transport by redistributing the non-equilibrium phonon modes in the phase space.35,36 Isotope scattering is also resistive to heat transport. Therefore, U process and isotope scattering together are called resistive (R) process in this paper.

In the 1960s, Guyer and Krumhansl37 proposed a criterion to determine hydrodynamic phonon transport by comparing the averaged phonon scattering rate as
Γ ¯ N Γ ¯ B and Γ ¯ N Γ ¯ R ,
(1)
where Γ ¯ N, Γ ¯ B, and Γ ¯ R are the averaged phonon scattering rates for the N process, boundary scattering, and R process, respectively. The extrinsic boundary scattering rate is defined by Casimir theory38 as Γ B = 2 | v | / d, where d is the sample width and v is the phonon group velocity perpendicular to the boundary. Equation (1) means that when the momentum-conserving N process is dominant, the phonons undergo hydrodynamic phonon transport in the form of collective excitation motion. Depending on the relative strength order between Γ ¯ B and Γ ¯ R, the hydrodynamic phonon transport can be further categorized into Poiseuille and Ziman regimes (see Fig. 1), which correspond to the boundary scattering and R process dissipated heat flow, respectively. At low temperature, when the phonon MFP is much larger than the sample size, the boundary scattering dominates in the scattering processes so that the phonons exhibit ballistic transport behavior without internal collision, as shown in Fig. 1. On the other hand, when the sample size is much larger than the phonon MFP, the R process is dominant in the phonon scatterings, giving rise to the kinetic (diffusive) thermal transport behavior that is typically observed in bulk crystals.
FIG. 1.

Schematic diagram of thermal conductivity κ vs temperature indicating different thermal transport regimes. Here, Γ ¯ B, Γ ¯ N, and Γ ¯ R represent the averaged scattering rate for the boundary scattering, the normal process, and the resistive process (including Umklapp scattering and isotopic scattering), respectively.

FIG. 1.

Schematic diagram of thermal conductivity κ vs temperature indicating different thermal transport regimes. Here, Γ ¯ B, Γ ¯ N, and Γ ¯ R represent the averaged scattering rate for the boundary scattering, the normal process, and the resistive process (including Umklapp scattering and isotopic scattering), respectively.

Close modal
Guyer's criterion remains the most widely used in literature studies, despite the existence of other criteria.39,40 In Guyer's criterion, the averaged scattering rate over various phonon frequencies is calculated as26,
Γ ¯ i = C ( ω ) Γ i ( ω ) C ( ω ) ,
(2)
where i represents either N process, U process, or isotopic scattering. Here, C ( ω ) = n 0 ( ω ) [ n 0 ( ω ) + 1 ] ( ω ) 2 / ( k B T 2 ) is the mode specific heat, where n 0 denotes the equilibrium Bose–Einstein distribution function, k B is the Boltzmann constant, is the reduced Planck constant, ω is the phonon frequency, and T is the temperature. A recent study by Zhang et al.41 found that the torsional mode located in the low-frequency region of bulk crystalline polyethylene caused a complete separation of the dominant peak spectrum between the mode specific heat C ( ω ) and mode thermal conductivity κ ( ω ). As a result, Eq. (2) cannot correctly predict the temperature at which the hydrodynamic phonon transport in bulk crystalline polyethylene occurs. In order to generalize the criterion, they further proposed to use κ ( ω ) as the weighting ratio in the calculation of averaged scattering rate as41 
Γ ¯ i = κ ( ω ) Γ i ( ω ) κ ( ω ) ,
(3)
which can accurately predict the emergence of hydrodynamic phonon transport in bulk crystalline polymers in good agreement with independent predictions from Boltzmann transport equation.41 

In semiconductor solids, phonons are the main heat carriers. The solid line in Fig. 1 shows the typical temperature-dependent thermal conductivity in solids. The ballistic transport dominates at low temperature due to the limited internal scatterings. With the increase of temperature, the phonons enter the hydrodynamics transport regime, and κ will increase due to the introduction of a large number of N processes. When the temperature is further increased, the R process becomes stronger and overtakes the boundary scattering, dominating the phonon transport in the Ziman regime. It is noteworthy that the N process is always much stronger than the other phonon processes in the hydrodynamic regime. This is also an important reason why materials show high κ in a certain temperature range. Interestingly, the temperature range of hydrodynamic phonon transport in 2D materials is higher than that in the three-dimensional (3D) bulk materials, which stimulates many recent research interests to study hydrodynamic phonon transport behavior based on 2D materials.

The micro-and nanoscale heat transport phenomenon can be theoretically described by macroscopic methods,42,43 which can obtain the heat transport property of the system based on the macroscopic physical quantity simply and efficiently. Numerical models43,44 for hydrodynamic phonon transport is a relatively strict statistical physical model among various macroscopic methods since it can be directly derived from the phonon Boltzmann equation.

Due to complex scattering terms, it is difficult to obtain a strict analytical solution of the phonon Boltzmann equation. The Callaway dual relaxation model45 simplifies the scattering terms and considers the N and U phonon scattering separately. Under the Callaway dual relaxation model, the phonon Boltzmann equation is expressed as45,
n t + v g n = n n N e q τ N n n R e q τ R ,
(4)
where τ N and τ R are the relaxation time for the N process and R process (including both U process and isotopic scattering), respectively, and v g is the phonon group velocity. The R process causes phonon distribution to approach the Bose–Einstein distribution46  n R e q = 1 exp ( ω k B T ) 1, whereas the N process causes phonon distribution to approach the displaced Bose–Einstein distribution47  n N e q = 1 exp ( ω k u k B T ) 1, in which u is the drift velocity describing the collective phonon motion and k is the phonon wavevector.

In the exploration of hydrodynamic phonon transport, various phonon hydrodynamic equations (e.g., Fourier's heat equation,48 Cattaneo–Vernotte heat equation,49,50 and Guyer -Krumhansl (G -K) heat equation43) obtained by an approximate solution of the phonon Boltzmann equation have been developed with different methods, such as the Chapman–Enskog expansion method,51–53 the four-field equations method,54,55 the maximum entropy moment method,56–58 Grad's type moment method,55,59,60 and the eigenvalue analysis method.37,43,61 Among various numerical models, the G-K heat equation based on the Callaway dual relaxation model45 is considered to be capable of accurately describing the heat transport behavior in the hydrodynamic region. The G-K heat equation was first derived by the eigenvalue analysis method37,43 and then obtained by other methods.52–55 Initially, the G-K heat equation was valid only for 3D materials at low temperatures, which was further developed to work at room temperature or high temperature.62,63 In addition, there also exist other schemes for solving the phonon Boltzmann equation, including the Monte Carlo scheme,64 the lattice Boltzmann method,48 the discrete-ordinate-method scheme,65,66 the finite volume scheme,67 the discrete unified gas kinetic scheme,68 and the iterative approach,69 

The classical Fourier law based on the widely used single-mode relaxation approximation fails in hydrodynamics because it incorrectly treats the N process as resistive as the same as the U process. As a result, in the hydrodynamic regime when the N process is dominant, the difference of the computed κ between the single-mode relaxation approximation and the exact solution by the iterative approach will become significant, serving as a signal to indirectly probe the hydrodynamic phonon transport.41 

It is worth noting that the above methods are derived from 3D materials. Due to the existence of the flexural acoustic (ZA) mode in 2D materials, modified hydrodynamic models for 2D materials have been developed in recent years.32,66,70,71 Furthermore, as the classical hydrodynamic phonon theory is applicable only at low temperatures, the emergence of hydrodynamics at high temperature for 2D materials brings new challenges. In order to establish the hydrodynamic phonon equation at room temperature or high temperature, many efforts have been made on the basis of the G-K heat equation.62,63,72–76

The study of phonon hydrodynamics originated from the exploration of thermal wave phenomena at low temperature in bulk dielectric solids.77 Guyer and Krumhansl37,43 used the eigenvalue analysis method to numerically solve the phonon Boltzmann equation and obtained the G-K heat equation. Subsequent mathematicians and physicists have carried out many research studies on the theoretical solution of phonon hydrodynamics.62,63,72–76

Because the traditional method for solving the phonon Boltzmann equation is incomplete,42 Michel et al.31 derived the coupled integrodifferential equations for the in-plane acoustic displacement correlations and ZA mode density fluctuations in 2D crystals. By applying the linear response theory and thermal Green's functions, they found that the fluctuation of phonon density distribution causes thermal stress in the sound wave equation, while in-plane lattice displacement acts as a perturbing field in the kinetic equation. This is because the modulation of the ZA mode by the in-plane sound wave results in an out-of-equilibrium density distribution of the ZA mode, and the perturbed density distribution in turn influences the sound wave equation due to the thermo-mechanical coupling. These coupled dynamic equations have enabled the study of 2D phonon hydrodynamics on thermal–mechanical effects.33,71

Afterward, Scuracchio et al.71 derived the corresponding hydrodynamic equations considering two situations where one is energy conservation only and the other is energy and crystal momentum conservation. They found that thermal–mechanical coupling induced thermal resonances (Landau–Placzek peak and second sound) appear in the displacement susceptibility, while the acoustic sound wave doublet appears in the temperature susceptibility. The influence of thermal–mechanical coupling is reflected in the recent transient thermal grating experiment33 for the observation of the second sound phenomenon.

To develop the numerical solution of the phonon Boltzmann equation under the Callaway dual relaxation model,45 Guo and Wang66 developed the discrete-ordinate-method scheme to study heat transport in 2D materials. Their scheme provides a good approximation to the ab initio calculation and is capable of modeling different geometries. Taking graphene as an example, they demonstrated that their developed scheme can produce results consistent with the existing results from molecular dynamics simulations, Monte Carlo simulations, and experimental measurements. They further predicted that the phonon Knudsen minimum phenomenon46 in a graphene ribbon can exist only at low temperature and low isotope concentration. The improvement of the traditional method for solving the Boltzmann transport equation provides useful tools to the study of hydrodynamic phonon transport in 2D materials.

Phonon hydrodynamics can lead to various peculiar phenomena, such as second sound,37 Poiseuille flow,37,78 propagation and attenuation of sound waves,79 and ultrahigh κ.26 Among them, the second sound, also known as the thermal wave, is related to the propagation of temperature waves caused by thermal pulses.77 It should be pointed out that the thermal wave phenomenon is different from the wave nature of phonons. Thermal wave represents the collective motion of phonons with different polarizations and is a material property, which does not necessarily require periodic structures. On the other hand, the wave nature of phonons, which represents the coherence of individual phonons, typically emerges in periodic structures.19,80–82 When the coherent phonons interfere with each other in periodic structures, interesting features, such as total-transmission and total-reflection of individual phonons, can be achieved in the presence of interfaces.82 

The second sound is the counterpart of the conventional sound wave (the first sound), whose velocity is the speed of sound in a solid. The difference between second sound and the conventional sound wave is that second sound represents a phonon density wave in which the phonons exhibit a collective motion, while the conventional sound wave represents the phonons in the long-wavelength limit that transport ballistically. Yao and Cao83 used molecular dynamics simulations to compare the transport of the conventional sound wave and the second sound wave in graphene at an ambient temperature of 50 K. By calculating the wave velocities, they found that the velocity of second sound ( v II ) is related to the conventional sound velocity (vI) as v II = v I 3, which is consistent with the relationship derived in the 3D Debye model.84 

However, the 3D Debye model cannot accurately describe graphene due to the existence of a quadratic ZA mode. As shown in Fig. 2(a), Shang et al.32 found that v II in graphene is dependent on the temperature (solid line) after taking into account the quadratic phonon dispersion, which is completely different from the temperature-independent prediction by the traditional Debye model (dashed and dotted lines), which predicts v II = v I 2 and v II = v I 3 for 2D and 3D cases, respectively. At low temperature, low-frequency phonons dominated by the ZA mode contribute most to v II. The contribution of the ZA mode becomes smaller as the temperature increases, leading to the smaller difference in v II between the Debye model and the model considering the quadratic ZA mode at higher temperature. Moreover, Lee et al.25 computed v II from the phonon dispersion obtained by first-principles calculations instead of using the Debye model. They found that the value of v II is almost the same as the group velocity of the ZA mode. Therefore, similar to the temperature-dependent group velocity of the ZA mode, v II increases with temperature. Using numerical simulations, Luo et al.63 found that the input thermal pulse first turns into two ballistic peaks, followed by the peak of the second sound [Fig. 2(b)]. Their results reveal that the velocity of two ballistic peaks are very close to the group velocity of LA and TA modes, while v II is almost contributed by the ZA mode.

FIG. 2.

(a) Theoretical prediction for the velocity of second sound v II vs temperature in a graphene nanoribbon. The dashed and dotted lines correspond to the 2D Debye model ( v II = v I 2 ) and 3D Debye model ( v II = v I 3 ), respectively. The solid line represents the model with one quadratic ZA mode and two linear acoustic modes. Reproduced with permission from Shang et al., Sci. Rep. 10, 8272 (2020). Copyright 2020 Springer Nature. (b) Local temperature distribution in an infinitely wide graphene ribbon without isotope scatterings at different times. Reproduced with permission from Luo et al., Phys. Rev. B 100, 155401 (2019). Copyright 2019 American Physical Society.

FIG. 2.

(a) Theoretical prediction for the velocity of second sound v II vs temperature in a graphene nanoribbon. The dashed and dotted lines correspond to the 2D Debye model ( v II = v I 2 ) and 3D Debye model ( v II = v I 3 ), respectively. The solid line represents the model with one quadratic ZA mode and two linear acoustic modes. Reproduced with permission from Shang et al., Sci. Rep. 10, 8272 (2020). Copyright 2020 Springer Nature. (b) Local temperature distribution in an infinitely wide graphene ribbon without isotope scatterings at different times. Reproduced with permission from Luo et al., Phys. Rev. B 100, 155401 (2019). Copyright 2019 American Physical Society.

Close modal

As a typical 2D material, graphene3 has ultrahigh κ and the quadratic dispersion of ZA modes.27–30 There are extensive discussions in literature studies on the importance of the ZA phonon to the thermal conductivity of graphene. Klemens85 believed that the contribution of the ZA phonon is small due to the small group velocity and the large Grüneisen parameter. Based on the iterative solution of the Peierls–Boltzmann transport equation and the three-phonon scattering process, Lindsay et al.29 predicted that the ZA mode can contribute up to 75% to the total κ in suspended single-layer graphene at room temperature due to the mirror-symmetry induced selection rule that forbids a substantial amount of phonon scatterings involving the ZA phonon. When further considering the higher order phonon scattering process, Feng and Ruan86 found that the contribution of the ZA phonon in graphene reduces to 30% after taking into account the four-phonon scattering process, although their prediction of thermal conductivity value is later found to be underestimated due to the neglected temperature-dependent interatomic force constant.87 They found that the mirror-symmetry rule allows for significantly more four-phonon scatterings than three-phonon scatterings, leading to the unprecedentedly high four-phonon scattering rates for the ZA phonon. Interested readers can refer to Refs. 88 and 89 for a more comprehensive overview on the relative contribution of the ZA phonon to the thermal conductivity of graphene.

Furthermore, the quadratic dispersion of the ZA mode is responsible for the hydrodynamic phonon transport in graphene. The large anharmonicity and density of states of the ZA mode in graphene enhance the N process. As the momentum-conserving N process gets stronger, the collective hydrodynamic phonon transport becomes more apparent.

The hydrodynamic phonon transport regime in graphene is remarkably different from that of 3D materials, which has only been observed experimentally at extremely low temperature and within narrow temperature ranges.90–95 Recent studies25,26 have shown that hydrodynamic phonon transport can exist in graphene even at room temperature. Lee et al.25 systematically studied hydrodynamic phonon transport from collective phonon motion, Poiseuille flow, and second sound through first-principles calculations. The cause of hydrodynamic phonon transport phenomena can be attributed to the collective drift motion of phonons.25 As shown in Fig. 3(a), this point can be revealed by the same slope of the drift component for three acoustic modes, which means all three acoustic modes share the same drift velocity. Since the thermal resistance in the Poiseuille regime is mainly caused by boundary scattering, there exists a gradient of the drift velocity perpendicular to the heat flow direction in the Poiseuille regime.

FIG. 3.

(a) The normalized distribution deviation of the three acoustic modes in graphene along the x direction. (b) The sample width ranges of Poiseuille flow in graphene. (a) and (b) are reproduced with permission from Lee et al., Nat. Commun. 6, 6290 (2015). Copyright 2015 Springer Nature. (c) The window of second sound with the natural isotope content in thin graphite measured in experiment. The color scale is the ratio of the maximum value at the resonant peak to the minimum value below the resonant peak frequency. Reproduced from Huberman et al., Science 364, 375 (2019). (d) Temperature-dependent κ of thin graphite measured in the experiment. The thickness of the graphite varies from 8.5 to 580 μm. Reproduced from Machida et al., Science 367, 309 (2020).

FIG. 3.

(a) The normalized distribution deviation of the three acoustic modes in graphene along the x direction. (b) The sample width ranges of Poiseuille flow in graphene. (a) and (b) are reproduced with permission from Lee et al., Nat. Commun. 6, 6290 (2015). Copyright 2015 Springer Nature. (c) The window of second sound with the natural isotope content in thin graphite measured in experiment. The color scale is the ratio of the maximum value at the resonant peak to the minimum value below the resonant peak frequency. Reproduced from Huberman et al., Science 364, 375 (2019). (d) Temperature-dependent κ of thin graphite measured in the experiment. The thickness of the graphite varies from 8.5 to 580 μm. Reproduced from Machida et al., Science 367, 309 (2020).

Close modal

Moreover, Lee et al.25 found that the κ in the hydrodynamic phonon transport regime depends on the sample size due to the external momentum loss mechanism, which is different from the size effect in the ballistic regime because the size in the hydrodynamic phonon regime is not much smaller than the MFP. Besides, a sufficiently large size may lead to the diffusive phonon transport. In Fig. 3(b), the upper and lower limits are set for the ballistic and diffusive transport, respectively, and hydrodynamic phonon transport will occur in between the upper and lower limits. Compared to the case of 3D materials (diamond), the phonon Poiseuille flow in suspended graphene can occur at higher temperatures and wider temperature ranges, which is due to the strong N process promoted by the ZA mode. Similarly, the second sound also has a large temperature window.25,33

Phonon hydrodynamics in 3D systems38,78,96 have been investigated for many years by experimental and theoretical studies, while the existing studies on phonon hydrodynamics in 2D materials are mostly theoretical, with very limited experimental progresses. Compared with the Poiseuille flow, the second sound is easier to be detected in experiments via the fluctuation of temperature. Recently, Huberman et al.33 directly observed the second sound heat transfer in 2D thin graphite with naturally occurring isotope impurities at temperatures above 100 K by using time-resolved optical measurements of heat transport on the micrometer scale. Their results show good agreements with ab initio calculations.25 As shown in Fig. 3(c), the second sound exists in a wide temperature range from 50 to 250 K. Phonon transport is ballistic at low temperature and small grating periods, while phonon transport is diffusive at high temperature and large grating periods.25 This classification of phonon transport regimes based on frequency-domain Green's functions is consistent with Fig. 1. Furthermore, the second sound can even emerge at room temperature with the decrease of the isotopic content or the layer thickness, or the increase of the grain size.97 

In order to explore the physical mechanism affecting the high κ of graphite, by monitoring the evolution of the κ in 2D thin graphite with temperature and thickness, Machida et al.34 studied the link between these two factors with phonon hydrodynamics resulting from phonon dispersion anisotropy. The experiment results34 in Fig. 3(d) show that κ of 2D thin graphite increases as the thickness decreases, which is due to the strong phonon boundary scattering effects. When the sample is thinner, a fraction of the U process is replaced by the specular boundary reflection, so hydrodynamic phonon transport that limits the degradation of the heat flow is promoted and κ increases.

To date, most studies on phonon hydrodynamics of 2D materials have been mainly based on graphene or 2D thin graphite sheets. Due to the similar properties of graphene and other 2D materials with high κ such as the hexagonal boron nitride and borophene, many theoretical26,31,66,71 and experimental33 studies on graphene can be generalized to other 2D materials.

In this paper, hydrodynamic phonon transport in 2D materials is discussed. First, we summarize the criterion and numerical methods for studying phonon hydrodynamic transport. Then, through reviewing the state-of-the-art works on hydrodynamics in 2D materials, a close correlation between phonon hydrodynamics and the ZA mode is determined.

In phonon hydrodynamics, the second sound intuitively reflects the collective excitation of phonons. Although the phenomenon of second sound has been observed in 2D materials, further studies are needed to characterize the second sound, especially the relationship between the ZA mode and second sound in 2D materials. Furthermore, as the second sound is very sensitive to the environment, effective approaches to remove the ambient noise in the experiment or simulation are very important. In addition, the influence of the external parameters that typically appear in experimental samples (e.g., strain, isotope, defect, and roughness) also needs to be taken into account.

The study on phonon hydrodynamics in 2D materials is still in its infancy, and most of the existing studies only focus on graphene. Since the family of 2D materials is quite large, further studies on the phonon hydrodynamics in other 2D materials should be conducted in order to establish a general relationship between the thermal conductivity and phonon hydrodynamics in 2D systems.

This work was supported in part by grants from the National Natural Science Foundation of China (Grant Nos. 12075168 and 11890703) and the Science and Technology Commission of Shanghai Municipality (Grant Nos. 19ZR1478600 and 18JC1410900).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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