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Theoretical and Applied Mechanics Letters

journal homepage: www.elsevier.com/locate/taml

Review

Heat transport in low-dimensional materials: A review and perspective

Zhiping Xu

Applied Mechanics Laboratory, Department of Engineering Mechanics and Center for Nano and Micro Mechanics, Tsinghua University, Beijing 100084, China

State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

CrossMark

highlights

• Heat transport is a key energetic process in materials and devices.

• The rich spectrum of imperfections in low-dimensional materials introduce fruitful phenomena.

• We focus on the roles of defects, disorder, interfaces, and the quantum-mechanical effect.

• New physics from theory and experiments is reviewed, followed by a perspective on open challenges.

article info

abstract

Article history:

Received 30 March 2016

Received in revised form

26 April 2016

Accepted 28 April 2016

Available online 9 May 2016

*This article belongs to the Solid Mechanics

Keywords:

Nanoscale heat transport

Low-dimensional materials

Defects

Disorder

Interfaces

Quantum mechanical effects

Heat transport is a key energetic process in materials and devices. The reduced sample size, low dimension of the problem and the rich spectrum of material imperfections introduce fruitful phenomena at nanoscale. In this review, we summarize recent progresses in the understanding of heat transport process in low-dimensional materials, with focus on the roles of defects, disorder, interfaces, and the quantum-mechanical effect. New physics uncovered from computational simulations, experimental studies, and predictable models will be reviewed, followed by a perspective on open challenges.

© 2016 The Author. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://

creativecommons.org/licenses/by-nc-nd/4.0/).

Contents

1. Introduction to thermal energy transport........................................................................................................................................................................113

2. Defects and effective medium theory...............................................................................................................................................................................115

3. Disorders and regime shift................................................................................................................................................................................................116

4. Interfacial thermal transport.............................................................................................................................................................................................117

5. Quantum mechanical effects.............................................................................................................................................................................................118

6. Perspectives........................................................................................................................................................................................................................119

Acknowledgments.............................................................................................................................................................................................................119

References...........................................................................................................................................................................................................................120

1. Introduction to thermal energy transport

* Correspondence to: Applied Mechanics Laboratory, Department of Engineering Mechanics and Center for Nano and Micro Mechanics, Tsinghua University, Beijing 100084, China.

E-mail address: xuzp@tsinghua.edu.cn.

Nanoscale heat transport is a key energetics process for the functioning and stability of integrated nanosystems and nanos-tructured materials, which hold great promises in a variety of applications ranging from energy management, conversion [1], phononics based computation [2], and thermotherapy for cells and tissues [3]. The nature of thermal energy transport is the

http://dx.doi.org/10.1016/j.taml.2016.04.002

2095-0349/© 2016 The Author. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article underthe CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Fig. 1. (a) Temperature evolution as a heat pulse propagates along a 1 ^m-long single-walled (5, 5) carbon nanotube (CNT), obtained from classical molecular dynamics simulations. The dash line indicates fronts of the ballistic longitudinal wave and collective heat wave, respectively. The velocities are vs and vh = vs/^J3. (b) Temperature dependence of thermal conductivity for 3D and 2D solids [4].

redistribution of kinetic energies in materials, toward thermal equilibrium or steady state under temperature gradient. This process demonstrates a ballistic behavior at short length scales, where energy propagation proceeds in forms of coherent waves (Fig. 1(a)). At another limit where the lattice vibration waves are strongly perturbed by scattering sources, diffusive behaviors are expected.

Between these two extremes there are mechanistic transitions taking place at specific length and time scales, or a critical concentration of resistive sources in the material. One feasible approach to formalize the intermediate regime is to divide the heat flux into two components in the transport processes, known as the ballistic-diffusive model [5,6]. One originates from the thermal boundaries and represents the ballistic part. The other component is contributed by scattered and excited carriers as diffusive processes. To characterize the transition in between, a mean free path /MFP can be defined for the carriers, below which a wave-like behavior is preserved. For crystalline nanostructures such as carbon and boron-nitride nanotubes, the phonon mean free path could reach hundreds of nanometers, which exceeds the characteristic length of structural ripples and approaches the theoretical limit set by their radius of curvature [7]. The robustness of heat conduction in these nanostructures refines the ultimate limit far beyond the reach of ordinary materials. In the ballistic limit, coherent wave propagation can be outlined by the Landauer formula through a transmission function defined for both elastic or inelastic scattering processes [8]. Beyond the length scale of ~/MFP, a thermal excitation emitted at the heat source loses its memory of both phase and momentum, and this diffusive nature allows a phenomenological description based on the diffusive Fourier's law. However, the parabolic diffusion equation predicts that a pulse of heat at the origin is felt at any distant point instantaneously, admitting an infinite speed of propagation of heat signals, which is in contradiction with the theory of relativity. To recover part of the wave-like nature into the diffusive picture and approach the ballistic limit, a number of updated models have been proposed by adding characteristic time scales in a hyperbolic form of equation, for example. Although being successful in solving some practical problems, works by Cattaneo [9], Vernotte [10], and Chester [11] following this spirit have been criticized for their phenomenological treatment that leads to both conceptual ambiguity in the definition of temperature and unrealistic behaviors in their predictions. Additional parameters have to be included for the time delay between heat flux and the temperature gradient in the fast-transient processes.

For crystalline solids where the notion of phonon is well defined, one can perform lattice dynamics calculations to determine harmonic and anharmonic force constants. These parameters can then be used to construct a semi-classical theory based on the phonon Boltzmann-Peierls transport equation (phBPE),

which is able to bridge the ballistic and diffusive models coherently [12]. The thermal conductivity of materials, k, could be predicted within this framework by including multiple-phonon scattering effects through high-order force constants, which can be calculated from density-functional theory based first-principles calculations. For example, the predicted values of k for silicon agree well with experimental measurements in a wide range of temperature [13,14]. This approach has also been validated for a wide spectrum of materials including those with complex structures such as the clathrate [15]. Moreover, this first-principles approach using phonon as the propagating quanta also yields detailed information of the heat transport process, such as mode-resolved relaxation time and resistance, and the resistive contribution from phonon-phonon interactions at different orders [14,16,17]. By decomposing the thermal resistance from individual scattering events including normal, resistive (Umklapp, isotopic) and extrinsic processes, classification of thermal transport regimes can be made. The Umklapp and isotopic processes are resistive, while the normal process can result in energy flow between phonon modes. The temperature dependence of thermal conductivity serves commonly as a spectroscopy to extract information of these scattering sources, and the resistive contribution from them can be added up using the Matthiessen's rule by assuming they are independent. Recent studies show that the dimension of materials plays a key role in the selection of regimes [4,18]. In graphene, as an example of two-dimensional (2D) crystals, normal processes dominate over the Umklapp scattering well-above the cryogenic condition, extending to room temperature and even more, which allows the Poiseuille and Ziman hydrodynamic regimes emerging at ordinary conditions, as demonstrated through drift motion of phonons, significance of boundary scattering and second sound (Fig. 1(b)) [4,18]. These facts indicate that the low-dimensional nature grants these nanostructures uncommon significance as model materials for the study of heat transport in the non-diffusive manner.

For material samples measured in thermal transport experiments, imperfections inevitably present at large length and time scales, which not only introduce thermal resistance, but may also alter the mechanism of heat transport. Defects, disorders, and interfaces in materials lead to weak or strong wave scatterings, and even localization of wave propagation. Scaling theory predicts that wave localization is more significant in lower dimensions [19], although compared to quantized waves such as the electrons, the spatial extension of vibrational waves, especially low-frequency waves at the continuum limit, is much larger and this effect is less significant [20]. These fundamental mechanisms of heat conduction considering these factors have not been well understood yet because of the lack of efficient control and characterization tools in exploring microscopic heat transport and dissipation processes in materials. The mean free path of phonons in nanostruc-tures such as CNTs or graphene is expected to be extraordinarily

Fig. 2. (a) Atomic heat flux in a single-crystalline graphene. (b) Defect scattering of heat flux by a single grain boundary in a graphene bi-crystal. (c), (d) Scattering by grain boundaries in polycrystalline graphene with grain sizes of 1 nm and 5 nm, respectively. (e) The dependence of thermal conductivity of polycrystalline graphene, plotted as a function of the grain size [34].

long due to the strong carbon bonds and low-dimensional confinement of phonons. Their low-dimensional nature also allows structural tailoring at nanoscale, for example, the creation and manipulation of material imperfections and interfaces [21]. These low-dimensional nanostructures thus are excellent model materials to probe the detailed energetic processes at the atomic level. In addition to the bulk, there are also significant interests in unraveling the multiscale nature of thermal transport in assemblies of nanostructures with the presence of interfaces between them is of critical importance. For example, in contrast to the interface between bulk materials, phonons transmitted across interfaces between nanostructures could be very different from those propagating inside them, and thus the interfacial interaction becomes dominating in determining the conductance [22]. A predictable model bridging not only the nanometer and macroscopic length scales, but also the ballistic and diffusive nature, needs to be developed to understand collective behaviors of thermal energy transfer therein.

Theoretical and experimental efforts made to study these rich phenomena have been recently summarized in reviews and books [23-25]. In this article, we will take a different approach by highlighting a few critical points these studies lead to. In the following part of this review, we will discuss the nature of heat conduction in low-dimensional solids, with focus on its correlation with the microstructures of materials, which can be engineered using the cutting-edge nanotechnology. Effects of defect scattering and the effective medium theory (EMT) will be addressed at first, followed by the role of disorder in shifting the heat transport regime, and then the microscopic mechanism of interfacial thermal conduction. Quantum mechanical effects will also be covered, by demonstrating limitations in the classical treatment of heat carriers. The presentation will be closed by a perspective list of open challenges on this topic. Discussions were made based on our recent work, as well as materials collected from the literature.

2. Defects and effective medium theory

Thermal energy is a collection of lattice vibrations, and its propagation in dielectric or semiconducting solids manifests heat conduction. The microscopic process can be clearly seen by tracking the evolution of a heat pulse or phonon wave packet in the material [26], which may experience perturbation from imperfections such as vacancies [27], isotopic impurities [28], and grain boundaries [29], in addition to the intrinsic phonon-phonon interaction. These extrinsic scattering processes break down the coherency of

waves and become inelastic if there are internal degrees of freedom. To quantify the probability that a phonon incident upon an imperfection with a certain crystal moment will be scattered, one needs to determine the scattering cross-section (width) for point (line) imperfections, or reflection coefficient for planar ones. To this end, one can derive formal solutions using Green's function of the unperturbed problem from the interatomic interaction near the imperfections or use elastic continuum theory [30]. One can also visualize the scattering processes by performing phonon-imaging experiments where a heat pulse is optically excited and the thermal energy propagation is tracked [31], or running molecular dynamics (MD) simulations, where the heat flux can be determined from the dynamics of atoms as

J = (^Vi - ^Su)/V, (1)

where ei, vi, Si are the energy, velocity, stress of atom i, and V is the volume of system. Recently, an expression for the adiabatic energy flux from first-principles calculations was derived, which allows the calculation of quantum-mechanical energy densities and currents [32]. Full anharmonicity of the interatomic interactions is included naturally here, and this approach could be used to assess both transient and steady state of a non-equilibrium system where the distribution of heat flux can be analyzed. For example, phonon frequencies and lifetimes can be fitted through the spectral energy density (SED) analysis [33]. The phonon lifetime or linewidth characterizes the interaction strength of a phonon with other quasi-particles, in particular charge carriers as well as its companion phonons. Using these techniques, we explored the scattering of heat flux in graphene by a single vacancy or Stone-Wales defect [27]. The results show that in a perfect lattice, the heat flux is weakly perturbed by phonon-phonon interactions with fluctuation in both the amplitude and direction, while, with the presence of point defects in the 2D lattice, strong scattering of the heat flow is observed. Similar observation has been reported in a following study of polycrystalline graphene, where the dislocations and grain boundaries act as scattering sources (Fig. 2) [34].

With a temperature difference set up between the heat source and sink, or a persistent heat flux is maintained across the sample, the steady state could eventually be approached in MD simulations. The relation between heat flux and the temperature gradient can be used to extract the "effective" thermal conductivity k along the temperature gradient from the Fourier's law

J = V TA, (2)

where A is the cross-section area of the sample. The thermal conductivity tensor k can also be calculated from the Green-Kubo

formula that relates correlation in the heat flux operator and

k = vklT^J a (T) •J (0)> dT' (3)

Although these two techniques have been widely used to measure k of nanostructures, their interfaces or bulk materials based on a supercell approach [23,24], one should keep in mind that they are derived based on a diffusive scenario of heat conduction, that is to say, the sample size should be much larger than lMFP. At mesoscopic scale with the sample size smaller than the characteristic length scale for inelastic scattering events, the preservation of wave characteristics such as interference and phase memory remains. The absence of inelastic scattering implies the disregard of dissipation and the concept of thermal conductivity is thus not well defined. Consequently, the effective thermal conductivity extracted from Eqs. (2) and (3) may include hidden parameters that are related to the microscopic details. In stark contrast, Landauer proposed a two-step view of the heat conduction process at mesoscopic scale including tunneling of waves and energy dissipation [35]. The nonlocal Landauer's formula predicts a wave transmission probability that is proportional to the conductance, and the element of dissipation exists at the contacts with thermal leads.

In practice of material or device design, usually one needs to predict the overall thermal conductivity of materials with the knowledge of defect concentration and distribution. This can be done by developing EMTs where defects are considered as inclusions in a composite [36]. The key parameters for these models are the thermal conductivities of components and their volume ratios. In applying this approach to microscopic phases, both of these two parameters should be carefully quantified, which have been reported to depend on the nature of defect and demonstrate concentration dependence [27,34,37]. For inclusions with size larger than that of atomic or molecular defects but on the order or smaller than the phonon mean path, the thermal conductivity of the inclusion can be well characterized, however, an interfacial or Kapitza resistance has to be considered in the EMT [36]. Moreover, the size of defects is usually not well defined at molecular level, and actually relies on the type of defects [27,34]. In a recent study, we reported that oxygen plasma treatment could reduce the thermal conductivity of graphene significantly even at extremely low defect concentration (~83% reduction for ~0.1% defects), which could be attributed mainly to the creation of carbonyl pair defects. Other types of defects such as hydroxyl, epoxy groups and nano-holes demonstrate much weaker effects on the reduction where the sp2 nature of graphene is better preserved [37]. By revealing this underlying correlation between thermal conductivity of the material with the defects it contains one could not only make theoretical predictions, but also identify the type of defects from the experimental data [37].

3. Disorders and regime shift

Unraveling the correlation between thermal transport in solids and their microstructures has been a continuing effort undertaken from both fundamental and application standpoints. Models have been proposed to describe thermal conduction at the microscopic level, which can be classified into two major categories. Perfect crystals feature translational lattice symmetries and thus the language of phonons apply as we discussed above [12]. Phonon dispersion and interaction parameters extracted from the equilibrium crystal structure can be used to derive kinetic models. Predictive calculations can then be done based on the specific heat, phonon group velocities, and rates of scattering processes following the phBPE [12]. In the other extreme of models, thermally excited hops between high-energy localized vibrational modes are

adopted to describe thermal diffusion in glassy materials, in addition to the low-frequency propagating modes. These localized modes do not propagate for long range but still carry heat and have a finite thermal diffusivity, which can be determined by the vibrational density of states and transition rates between them [38]. Allen and Feldma [38] developed a theory for this disordered regime with the idea that the dominant scattering is correctly described by a harmonic Hamiltonian, which is, in principle, transformable into a one-body problem of decoupled oscillators. The thermal conductivity can then be exactly calculated by an analog of the Kubo-Greenwood formula for electrical conductivity of disordered metals where Anderson localization is correctly contained. In the intermediate regime between crystals and glasses, the mixed nature of propagating and localized modes lends itself to the complexity where both the phonon propagating and hopping descriptions may both fail, and characters of atomic vibrations in glasses can be classified into propagons, diffusons and locons as the notion of "phonon" cannot be defined [39]. Insights into the heat conduction process in materials with intermediate level of disorder are limited due to this complexity, and rely much on microscopic experimental characterization and atomistic computer simulations at a certain level of disorder.

In low-dimensional solids, disorders can be introduced by defect, molecular doping, or localized symmetry breaking through structural transitions [37,40], which allow a quantitative study of their effects. Heat transport can be localized due to the presence of disorder. Atomic-scale disorder in solids attenuates propagating waves within a few angstroms, leading to localization of heat flux, while preserving the structural stability and stiffness of materials compared to porous materials [41,42]. However, in contrast to electronic transport, phonons have additional complexities, notably a broadband nature and strong temperature dependence. The phase information carried by high-frequency phonons can be lost through diffuse scattering at material boundaries and interfaces, low-frequency phonons in the continuum limit remain as coherent during their transport, for example, through the superlattice structures as reported recently [43,44].

The roles of disorder and heat flux localization in defining thermal properties of materials are more significant in low-dimensional materials than in their bulk counterparts [19,20]. The synthesis and characterization of 2D materials have seen remarkable advances recently. Experimental evidences have demonstrated the existence of crystalline, amorphous regions and their interfaces in silica bilayers and graphene. Even their growth and shrinkage could be tuned and tracked under control [45-49].These 2D materials thus provide an ideal platform to explore the effect of disorder in thermal conduction [50,51]. To gain insights into the thermal transport in 2D materials with varying levels of disorder, we performed MD simulations to analyze the results through theoretical models of heat transport in crystals and glasses [52]. We found that as the level of disorder a increases, the temperature dependence of thermal conductivity is turned over, from the signature behavior of crystal to that of the glass. The critical level of disorder is shown to correspond to the dominance of thermal diffusion from local mode hops over long-range phonon propagation, and the occurrence of heat flux localization.

Specifically, MD simulation results (Fig. 3) show that thermal transport in 2D silica with Stone-Wales type of disorder becomes dominated by the thermal hopping between localized modes with a > acr = 0.3, which explains a turnover observed in the temperature dependence of the thermal conductivity. The thermal conductivity decreases with temperature at a < acr, and increases at a > acr, and the T -dependence is weakened as a approaches acr for both cases. The determination of the critical level of disorder is validated from both the analysis of T -dependence turnover, dominance of diffusive contribution of thermal conductivity, the

participation ratio of localized modes and the spatial localization of heat flux, which all can be used as good indicators for the transition of heat transport mechanism. The localization of heat flux at high level of disorder results in dramatic reduction in the thermal conductivity [53]. This conclusion is expected to be general for other 2D and even bulk materials [27,37].

There are a few theoretical predictable models for heat transport that work well for the limiting cases of crystals and glasses, by introducing the microscopic mechanisms of propagating phonons or thermal hopping between localized modes. However, the applicability of these two classes of models need to be justified for materials with intermediate level of disorder, where either the polarization and group velocity of localized modes or the hopping and diffusivity of extended propagating modes are not well defined [54], but still make contributions to the heat conduction. To explore the detailed microscale dynamics related to this process of thermal energy transfer, the recently isolated 2D materials including a wide spectrum of crystalline lattices and types of disorder or defects provide an ideal test-bed.

4. Interfacial thermal transport

In conventional models of interfacial thermal transport, the interface is usually considered as a plane between two semiinfinite bulk matters and the information of interface itself is absent [55]. Consequently, the heat carriers propagating across the interface are assumed to have the same characters as those in the bulk, in order to predict the interfacial thermal conductance. The acoustic mismatch model (AMM) presumes that all phonons are governed by continuum acoustics and there is no scattering occurring at the interface. The transmission probability of phonon energy is calculated from the acoustic impedance. In the opposite extreme with strong diffusion at the interface, the diffusive mismatch model (DMM) replaces the complete secularity in AMM with the assumption that all of the phonons are diffusely scattered. The transmission is determined by a mismatch between the phonon densities of states. As reviewed in Ref. [55], the AMM and DMM set the upper and lower bounds of interfacial thermal conductance (ITC).

With the disregard of interfacial properties, these models could be problematic if the vibrational coupling across the interface is manifested by processes that are different from those contributing to the heat conduction in the materials. An apparent illustration is that the transmission coefficient between two weakly interacted identical nanostructures will be incorrectly predicted to be unit. Recent advances in exploring nanoscale interfaces have revealed critical roles of the interfacial structures and interactions in defining the heat transport process across them. A number of experiment evidences reported in the literature demonstrate a link between interfacial bonding characters and thermal conductance

at hard/soft and metal/dielectric interfaces [56-58]. This can also be clearly identified from the correlation between ITC for a wide spectrum of nanoscale interfaces and the interfacial energy, which are summarized in Fig. 4. The strong scattering at these weakly interacted interfaces permits the use of thermal resistance models in the interpretation of experimental or simulation results with the ITC [59-61].

The presence of interface for a material to others may lead to leaking of propagating phonons across it and thus reduces the heat flux in it. For example, strong interface scattering of flexural modes lowers the in-plane thermal conductivity of supported graphene or graphene multilayers in comparison to suspended graphene [75,76]. However, this reduction may compete with other resistive mechanisms such as the scattering from surfaces. A recent work shows that the thermal conductivity of a bundle of boron nanoribbons can be significantly higher than that of the freestanding one [77]. Moreover, the thermal conductivity of the bundle can be switched between the enhanced values and that of a single nanoribbon by wetting the van der Waals interface between the nanoribbons with various solutions [77]. For rough surfaces, the presence of free surface and nanoscale constriction is also critical for the heat transport. The pressure dependence of thermal conductance across contacts of rough surfaces was reported [78]. Their results were analyzed through a quantized phonon transmission model, where the pressure-dependent conductance per atom-atom contact is Gatom = GqNt. Here Gq = uk^T/6h is the universal quantum conductance that sets the upper limit for flow of heat and information across a single transport channel [8], N is the number of phonon modes and t is the transmission coefficients.

In nanoscale device applications where nanostructures are interfaced with other materials, efficient heat dissipation is necessary to maintain the performance and stability of nanoelectronic devices [61,79]. Due to the limited size of contact, the electrodes contribute insignificantly as heat sinks, while at least 84% of the electrical power supplied to the CNT electronics is dissipated directly into the substrate [80]. A recent study shows that interfacial intercalation could insulate the electronic coupling at the interface, while the ITC is not significantly reduced [81]. As a result, an electrically insulating but thermally transparent interfaces could be designed. Moreover, for these non-bonding interfaces between nanostructures such as CNTs, graphene, and other materials, transverse phonon modes are primarily responsible for thermal coupling, which are different from the heat carriers in the nanostruc-tures [82,83]. For example, MD simulation results show that for a CNT on a SiO2 substrate, although inelastic scattering between the CNT and substrate phonons contributes to the ITC, the thermal coupling is dominated by long-wavelength phonons between 0 and 10 THz. The energy transfer rate is much higher for the traverse optical (TO) modes than for the longitudinal acoustic (LA) and twist (TW) modes, while high-frequency (40-57 THz) CNT phonon

Fig. 4. The correlation between interfacial thermal conductance and interfacial energy [62]. Data is collected from the following references: CNT/sodium-dodecyl-sulfate (SDS) [63], Pt/water. [64], graphene/oil [65], graphene/phenolic resin [66], graphene/polyethylene (PE) [65], CNT/Pt [67], graphite/metal [68,69], diamond/Ti [70], graphene/SiO2 [71], graphene/SiC [66], copper/SiO2 [72], graphene/Cu(Ni) [73], benzene junction/diamond [61], TiN/MgO [74], and Au/self-assembled monolayer (SAM)/quartz [56].

Frequency (

Fig. 5. Vibrational spectra of carbyne, graphene, and diamond at 20 K, 100 K, and 300 K, obtained from path-integral molecular dynamics simulations [61].

modes are strongly coupled to sub-40 THz modes [82]. This mode-specific contribution to the ITC is different from the mechanism of heat conduction in the bulk and thus awaits an updated model for the prediction of ITC.

5. Quantum mechanical effects

In the classical treatment of atomic or ionic dynamics, all vibrational modes have the same energy of kBT according to the equipartition theorem. The quantum nuclear effect (QNE) that is critical at low temperature or for light elements is excluded. The Debye frequency wd defines a scale of temperature TD = hwD/kB below which the QNE becomes prominent. One can also roughly estimate the characteristic temperature Tq ~ hw for the rise of QNE significance at T < Tq, where w is frequency

of the vibrational mode. Figure 5 shows the vibrational spectrum of carbyne, graphene, and diamond, with the QNE implemented in path-integral MD simulations [61]. The data shows that as the quantum mechanical effects become more prominent as the dimension of materials is reduced, reflecting the increasing order of Debye temperature, that is TD = 1860 K for diamond, 2100 K for graphene, and 2800 K for carbyne, respectively [84,85].

In exploring thermal transport, the most significant quantum effects include the quantum statistics and the account ofzero point energy (ZPE). One simple solution is to correct, or renormalize the specific heat predicted from classical MD simulations. However, it was pointed out that applying quantum corrections to classical predictions, with or without the ZPE, does not bring them into better agreement with the quantum predictions compared to the uncorrected classical values for crystalline silicon above tem-

peratures of 200 K. Mode-dependent phonon properties demonstrate that the thermal conductivity cannot be corrected on a system level for the QNE [86]. On the other hand, a semi-quantum mechanical treatment can be made, for example, through coupling with a quantum color-noise thermostat [87]. This approach, also known as the generalized Langevin equation (GLE) method, crosses over to the ballistic and classical regimes in the low and high temperature limits. Recently there are a number of efforts made to capture the quantum effects in low-dimensional thermal transport through steady-state non-equilibrium MD simulations [87-90], which show that at temperature well below the Debye temperature, e.g. T < 500 K for CNTs, the quantum correction to MD simulations is significant and classical results strongly overestimate the specific heat and thus thermal conductivity. However, coupling to a quantum heat bath does not solve the problem as the ZPE is represented by tangible vibrations [87].

The quantum statistical effect is included in models such as the phBPE and non-equilibrium Green's function (NEGF). However, the distribution function based phBPE approach lacks of microscopic dynamics information and requires expensive computations for the phonon frequencies and lifetimes, and the NEGF formalism is only able to calculate ballistic transport coefficients at low temperature where the lattice vibrations are not significant. With these facts, direct simulations of quantum-mechanical thermal conduction in materials still remain as an open challenge, although directly solving the Schrodinger equation [91], or adopting a pathintegral formalism for many-particle systems [92,93], may be able to include the QNE in a natural and feasible way.

6. Perspectives

With recent interests in developing integrated nanosystems and nanocomposites, the defects, surfaces, interfaces have been explored as routes to modulate thermal transport, dissipation and management in materials. Nevertheless, there are a number of critical issues that remain as open challenges. Phonons play a dominating role in thermal energy transport in dielectrics and intrinsic semi-conductors, as we discussed above. However, electrons and holes also act as heat carriers in doped semiconductors and metals. Based on first-principles calculations and the Boltzmann-Peierls equation (BPE) for phonons and electrons, a recent work reported mode-dependent phonon transport properties of metals, where both phonon-phonon and electron-phonon interactions are considered. The results match well with experimental data for non-magnetic metals and it is found that the contribution of phonons in thermal conduction increases in thin metal films [94]. Another calculation shows that the in-plane electronic thermal conductivity of doped graphene is ~300 W/mK at room temperature, independently of doping. This result is much larger than expected and comparable to the total thermal conductivity of typical metals, contributing ~10% to the total thermal conductivity of bulk graphene [95]. Empirically, a two-temperature model was constructed to include the electron degree of freedom for thermal conduction across metal-non-metal interfaces [96].

To characterize the electronic contribution to thermal conductivity, the Wiedemann-Franz (W-F) law states that the ratio between the electronic thermal conductivity Ke and the electrical conductivity a at a given temperature T is a constant, i.e. the Lorenz numberL = Ke/(aT) = (n2/3)(kB/e)2, which reflects the fact that thermal and electrical currents are carried by the same Fermionic quasiparticles [97]. The W-F law was firstly proposed for bulk materials and further extended to the interfacial conductance [98]. For the 2D crystal graphene, this law is broadly satisfied at low and high temperatures, but with the largest deviations of 20%-50% around room temperature [95], which in quasi-one-dimensional

conductors where conventional Fermi-liquid theory and the picture of quasi-particle break down, it was found to be violated [99]. A direct simulation of the energy transport process accounting for both lattice vibrations and electronic transport is thus needed, which could assess validity of the W-F law and predict, for example, thermoelectric transport processes and power dissipation in nanoelectronics [100].

In nanoelectronics that represent the ultimate limit to the miniaturization of electrical circuits, power dissipation remains poorly understood owing to experimental challenges just as the heat transport. Electrical currents in the device induces overpopulation of phonons, which may in turn impede transport and enhance scattering with electrons as a result of strong elec-tron-phonon interaction. Joule heating and heat dissipation are the fundamental processes for charge transport, which arise from the interaction between electronic charge carriers and molecular vibrations and pose serious stability issues in these devices. Recent works probed the local non-equilibrium between electronic and phononic temperatures in understanding the heat dissipation [101]. The heat dissipation in atomic-scale junctions is found to be intimately related to the transmission characteristics of the junctions as predicted by the Landauer theory [60]. A correlated electrical-thermal transport model was developed for graphene-on-insulator devices by introducing a coupled solution of the continuity, thermal and electrostatic equations, offering insights into the formation of hot spot in device, the carrier distributions, fields and power dissipation in the graphene field effect transistor [102]. In addition to conduction, direct energy transfer from hot conduction electrons to the substrate polar surface phonons (field coupling) needs also to be included, especially at elevated temperatures [103,104], and the nanoscale control of phonon excitation may further enrich this process [105].

There have been significant and rising interest in developing nano-devices for biomedical applications involving interfaces between biological tissues and functional nanostructures [106-111]. These nano-devices are able to probe biophysiochem-ical signals, collect valuable information from key biological processes, and modulate cellular activities [106-111], which need to be adjacent or even embedded in biological tissues. Nano-devices with high complexity in integration requires integrated power supply and thus generates heat localization at the extremely small length scale [112]. The nano-devices made of synthetic materials may have material properties of high contrast to biological tissues, which thus introduce a thermal barrier to apply heat cues to regulate living systems. The uncertainty about thermal management at the interface between tissues and nano-devices has resulted in a lot of concerns and the interfacial thermal energy transfer is no doubt pivotal in designing relevant nano-devices. Intentional thermal management in biological tissues has also been demonstrated as an effective control for gene express, tumor metabolism and cell-selective treatment for diseases [113-116]. Key questions that need to be addressed for related biomedical applications are how energy inputs lead to local temperature rise at such a nanoscale interface and what criterion is to activate the thermoregulation for living systems [117]. These concerns could be addressed by exploring the extreme interfaces between low-dimensional functional devices and cellular entities, by in vivo, in vitro, or in silico studies.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11222217) and the State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics (MCMS-0414G01). The computation was performed on the Explorer 100 cluster system at the Tsinghua National Laboratory for Information Science and Technology.

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