Theoretical and Applied Mechanics Letters 6 (2016) 113–121

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Theoretical and Applied Mechanics Letters

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Review

Heat transport in low-dimensional materials: A review andperspectiveZhiping Xu ∗

Applied Mechanics Laboratory, Department of Engineering Mechanics and Center for Nano and Micro Mechanics, Tsinghua University, Beijing 100084,ChinaState Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

h i g h l i g h t s

• 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.

a r t i c l e i n f o

Article history:Received 30 March 2016Received in revised form26 April 2016Accepted 28 April 2016Available online 9 May 2016*This article belongs to the Solid Mechanics

Keywords:Nanoscale heat transportLow-dimensional materialsDefectsDisorderInterfacesQuantum mechanical effects

a b s t r a c t

Heat transport is a key energetic process inmaterials and devices. The reduced sample size, lowdimensionof the problem and the rich spectrum of material imperfections introduce fruitful phenomena atnanoscale. In this review, we summarize recent progresses in the understanding of heat transport processin 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, andpredictable 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 andApplied 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 ........................................................................................................................................................................1132. Defects and effective medium theory...............................................................................................................................................................................1153. Disorders and regime shift ................................................................................................................................................................................................1164. Interfacial thermal transport.............................................................................................................................................................................................1175. Quantum mechanical effects .............................................................................................................................................................................................1186. Perspectives........................................................................................................................................................................................................................119

Acknowledgments .............................................................................................................................................................................................................119References...........................................................................................................................................................................................................................120

∗ Correspondence to: Applied Mechanics Laboratory, Department of EngineeringMechanics and Center for Nano and Micro Mechanics, Tsinghua University, Beijing100084, China.

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

http://dx.doi.org/10.1016/j.taml.2016.04.0022095-0349/© 2016 The Author. Published by Elsevier Ltd on behalf of The Chinese SocBY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction to thermal energy transport

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

iety of Theoretical and Applied Mechanics. This is an open access article under the CC

114 Z. Xu / Theoretical and Applied Mechanics Letters 6 (2016) 113–121

Fig. 1. (a) Temperature evolution as a heat pulse propagates along a 1 µm-long single-walled (5, 5) carbon nanotube (CNT), obtained from classical molecular dynamicssimulations. The dash line indicates fronts of the ballistic longitudinal wave and collective heat wave, respectively. The velocities are vs and vh = vs/

√3. (b) Temperature

dependence of thermal conductivity for 3D and 2D solids [4].

redistribution of kinetic energies in materials, toward thermalequilibrium or steady state under temperature gradient. This pro-cess demonstrates a ballistic behavior at short length scales, whereenergypropagationproceeds in formsof coherentwaves (Fig. 1(a)).At another limitwhere the lattice vibrationwaves are strongly per-turbed by scattering sources, diffusive behaviors are expected.

Between these two extremes there are mechanistic transitionstaking place at specific length and time scales, or a criticalconcentration of resistive sources in the material. One feasibleapproach to formalize the intermediate regime is to divide theheat flux into two components in the transport processes, knownas the ballistic–diffusive model [5,6]. One originates from thethermal boundaries and represents the ballistic part. The othercomponent is contributed by scattered and excited carriers asdiffusive processes. To characterize the transition in between, amean free path lMFP can be defined for the carriers, below whicha wave-like behavior is preserved. For crystalline nanostructuressuch as carbon and boron-nitride nanotubes, the phonon meanfree path could reach hundreds of nanometers, which exceedsthe characteristic length of structural ripples and approaches thetheoretical limit set by their radius of curvature [7]. The robustnessof heat conduction in these nanostructures refines the ultimatelimit far beyond the reach of ordinary materials. In the ballisticlimit, coherent wave propagation can be outlined by the Landauerformula through a transmission function defined for both elasticor inelastic scattering processes [8]. Beyond the length scale of∼lMFP, a thermal excitation emitted at the heat source loses itsmemory of both phase and momentum, and this diffusive natureallows a phenomenological description based on the diffusiveFourier’s law. However, the parabolic diffusion equation predictsthat a pulse of heat at the origin is felt at any distant pointinstantaneously, admitting an infinite speed of propagation of heatsignals, which is in contradiction with the theory of relativity. Torecover part of the wave-like nature into the diffusive picture andapproach the ballistic limit, a number of updated models havebeen proposed by adding characteristic time scales in a hyperbolicformof equation, for example. Although being successful in solvingsome practical problems, works by Cattaneo [9], Vernotte [10],and Chester [11] following this spirit have been criticized fortheir phenomenological treatment that leads to both conceptualambiguity in the definition of temperature and unrealisticbehaviors in their predictions. Additional parameters have to beincluded for the time delay between heat flux and the temperaturegradient in the fast-transient processes.

For crystalline solids where the notion of phonon is welldefined, one can perform lattice dynamics calculations to deter-mine harmonic and anharmonic force constants. These parame-ters can then be used to construct a semi-classical theory basedon the phonon Boltzmann–Peierls transport equation (phBPE),

which is able to bridge the ballistic and diffusive models coher-ently [12]. The thermal conductivity of materials, κ , could bepredicted within this framework by including multiple-phononscattering effects through high-order force constants, which canbe calculated from density-functional theory based first-principlescalculations. For example, the predicted values ofκ for silicon agreewell with experimental measurements in a wide range of temper-ature [13,14]. This approach has also been validated for a widespectrum of materials including those with complex structuressuch as the clathrate [15]. Moreover, this first-principles approachusing phonon as the propagating quanta also yields detailed in-formation of the heat transport process, such as mode-resolvedrelaxation time and resistance, and the resistive contribution fromphonon–phonon interactions at different orders [14,16,17]. Bydecomposing the thermal resistance from individual scatteringevents including normal, resistive (Umklapp, isotopic) and extrin-sic processes, classification of thermal transport regimes can bemade. The Umklapp and isotopic processes are resistive, while thenormal process can result in energy flow between phonon modes.The temperature dependence of thermal conductivity serves com-monly as a spectroscopy to extract information of these scatteringsources, and the resistive contribution from them can be addedup using the Matthiessen’s rule by assuming they are indepen-dent. Recent studies show that the dimension of materials playsa key role in the selection of regimes [4,18]. In graphene, as anexample of two-dimensional (2D) crystals, normal processes dom-inate over the Umklapp scatteringwell-above the cryogenic condi-tion, extending to room temperature and evenmore, which allowsthe Poiseuille and Ziman hydrodynamic regimes emerging at ordi-nary conditions, as demonstrated through driftmotion of phonons,significance of boundary scattering and second sound (Fig. 1(b))[4,18]. These facts indicate that the low-dimensional nature grantsthese nanostructures uncommon significance as model materialsfor the study of heat transport in the non-diffusive manner.

For material samples measured in thermal transport experi-ments, imperfections inevitably present at large length and timescales, which not only introduce thermal resistance, but may alsoalter the mechanism of heat transport. Defects, disorders, and in-terfaces in materials lead to weak or strong wave scatterings, andeven localization of wave propagation. Scaling theory predicts thatwave localization is more significant in lower dimensions [19], al-though compared to quantized waves such as the electrons, thespatial extension of vibrational waves, especially low-frequencywaves at the continuum limit, is much larger and this effect isless significant [20]. These fundamental mechanisms of heat con-duction considering these factors have not been well understoodyet because of the lack of efficient control and characterizationtools in exploring microscopic heat transport and dissipation pro-cesses in materials. The mean free path of phonons in nanostruc-tures such as CNTs or graphene is expected to be extraordinarily

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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 grainboundaries in polycrystalline graphene with grain sizes of 1 nm and 5 nm, respectively. (e) The dependence of thermal conductivity of polycrystalline graphene, plotted asa function of the grain size [34].

long due to the strong carbon bonds and low-dimensional confine-ment of phonons. Their low-dimensional nature also allows struc-tural tailoring at nanoscale, for example, the creation and manip-ulation of material imperfections and interfaces [21]. These low-dimensional nanostructures thus are excellent model materials toprobe the detailed energetic processes at the atomic level. In ad-dition to the bulk, there are also significant interests in unravel-ing the multiscale nature of thermal transport in assemblies ofnanostructures with the presence of interfaces between them isof critical importance. For example, in contrast to the interfacebetween bulk materials, phonons transmitted across interfacesbetween nanostructures could be very different from those propa-gating inside them, and thus the interfacial interaction becomesdominating in determining the conductance [22]. A predictablemodel bridging not only the nanometer and macroscopic lengthscales, but also the ballistic and diffusive nature, needs to be devel-oped to understand collective behaviors of thermal energy transfertherein.

Theoretical and experimental efforts made to study theserich phenomena have been recently summarized in reviews andbooks [23–25]. In this article, we will take a different approachby highlighting a few critical points these studies lead to. In thefollowing part of this review, we will discuss the nature of heatconduction in low-dimensional solids, with focus on its correlationwith the microstructures of materials, which can be engineeredusing the cutting-edge nanotechnology. Effects of defect scatteringand the effective medium theory (EMT) will be addressed at first,followed by the role of disorder in shifting the heat transportregime, and then themicroscopicmechanismof interfacial thermalconduction. Quantum mechanical effects will also be covered,by demonstrating limitations in the classical treatment of heatcarriers. The presentation will be closed by a perspective list ofopen challenges on this topic. Discussions weremade based on ourrecent 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 prop-agation in dielectric or semiconducting solids manifests heat con-duction. The microscopic process can be clearly seen by trackingthe evolution of a heat pulse or phonon wave packet in the mate-rial [26], which may experience perturbation from imperfectionssuch as vacancies [27], isotopic impurities [28], and grain bound-aries [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 free-dom. To quantify the probability that a phonon incident uponan imperfection with a certain crystal moment will be scattered,one needs to determine the scattering cross-section (width) forpoint (line) imperfections, or reflection coefficient for planar ones.To this end, one can derive formal solutions using Green’s func-tion of the unperturbed problem from the interatomic interactionnear the imperfections or use elastic continuum theory [30]. Onecan also visualize the scattering processes by performing phonon-imaging experiments where a heat pulse is optically excited andthe thermal energy propagation is tracked [31], or running molec-ular dynamics (MD) simulations, where the heat flux can be deter-mined from the dynamics of atoms as

J = (6ieivi − 6iSivi)/V , (1)

where ei, vi, Si are the energy, velocity, stress of atom i, and V isthe volume of system. Recently, an expression for the adiabatic en-ergy flux from first-principles calculations was derived, which al-lows the calculation of quantum-mechanical energy densities andcurrents [32]. Full anharmonicity of the interatomic interactions isincluded naturally here, and this approach could be used to assessboth transient and steady state of a non-equilibrium systemwherethe distribution of heat flux can be analyzed. For example, phononfrequencies and lifetimes can be fitted through the spectral energydensity (SED) analysis [33]. The phonon lifetime or linewidth char-acterizes the interaction strength of a phonon with other quasi-particles, in particular charge carriers as well as its companionphonons. Using these techniques, we explored the scattering ofheat flux in graphene by a single vacancy or Stone–Wales de-fect [27]. The results show that in a perfect lattice, the heat fluxis weakly perturbed by phonon–phonon interactions with fluctua-tion in both the amplitude and direction, while, with the presenceof point defects in the 2D lattice, strong scattering of the heat flowis observed. Similar observation has been reported in a followingstudy of polycrystalline graphene,where the dislocations and grainboundaries act as scattering sources (Fig. 2) [34].

With a temperature difference set up between the heat sourceand sink, or a persistent heat flux is maintained across thesample, the steady state could eventually be approached in MDsimulations. The relation between heat flux and the temperaturegradient can be used to extract the ‘‘effective’’ thermal conductivityκ along the temperature gradient from the Fourier’s law

J = ∇TA, (2)

where A is the cross-section area of the sample. The thermalconductivity tensor κ can also be calculated from the Green–Kubo

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formula that relates correlation in the heat flux operator and

κ =1

VkBT 2

⟨J (τ ) · J (0)⟩ dτ . (3)

Although these two techniques have been widely used to measureκ of nanostructures, their interfaces or bulk materials based on asupercell approach [23,24], one should keep in mind that they arederived based on a diffusive scenario of heat conduction, that is tosay, the sample size should bemuch larger than lMFP. Atmesoscopicscale with the sample size smaller than the characteristic lengthscale for inelastic scattering events, the preservation of wave char-acteristics such as interference and phase memory remains. Theabsence of inelastic scattering implies the disregard of dissipationand the concept of thermal conductivity is thus not well defined.Consequently, the effective thermal conductivity extracted fromEqs. (2) and (3) may include hidden parameters that are relatedto the microscopic details. In stark contrast, Landauer proposed atwo-step view of the heat conduction process at mesoscopic scaleincluding tunneling of waves and energy dissipation [35]. The non-local Landauer’s formula predicts a wave transmission probabilitythat is proportional to the conductance, and the element of dissi-pation exists at the contacts with thermal leads.

In practice of material or device design, usually one needs topredict the overall thermal conductivity of materials with theknowledge of defect concentration and distribution. This can bedone by developing EMTs where defects are considered as inclu-sions in a composite [36]. The key parameters for these models arethe thermal conductivities of components and their volume ratios.In applying this approach to microscopic phases, both of these twoparameters should be carefully quantified, which have been re-ported to depend on the nature of defect and demonstrate concen-tration dependence [27,34,37]. For inclusions with size larger thanthat of atomic ormolecular defects but on the order or smaller thanthe phonon mean path, the thermal conductivity of the inclusioncan be well characterized, however, an interfacial or Kapitza resis-tance has to be considered in the EMT [36]. Moreover, the size ofdefects is usually not well defined at molecular level, and actuallyrelies on the type of defects [27,34]. In a recent study, we reportedthat oxygen plasma treatment could reduce the thermal conduc-tivity of graphene significantly even at extremely low defect con-centration (∼83% reduction for ∼0.1% defects), which could beattributed mainly to the creation of carbonyl pair defects. Othertypes of defects such as hydroxyl, epoxy groups and nano-holesdemonstrate much weaker effects on the reduction where the sp2

nature of graphene is better preserved [37]. By revealing this un-derlying correlation between thermal conductivity of the materialwith the defects it contains one could not only make theoreticalpredictions, but also identify the type of defects from the experi-mental data [37].

3. Disorders and regime shift

Unraveling the correlation between thermal transport in solidsand their microstructures has been a continuing effort un-dertaken from both fundamental and application standpoints.Models have been proposed to describe thermal conduction at themicroscopic level, which can be classified into two major cate-gories. Perfect crystals feature translational lattice symmetries andthus the language of phonons apply as we discussed above [12].Phonon dispersion and interaction parameters extracted from theequilibrium crystal structure can be used to derive kinetic mod-els. Predictive calculations can then be done based on the specificheat, phonon group velocities, and rates of scattering processes fol-lowing the phBPE [12]. In the other extreme of models, thermallyexcited hops between high-energy localized vibrational modes are

adopted to describe thermal diffusion in glassy materials, in ad-dition to the low-frequency propagating modes. These localizedmodes do not propagate for long range but still carry heat and havea finite thermal diffusivity, which can be determined by the vibra-tional density of states and transition rates between them [38].Allen and Feldma [38] developed a theory for this disorderedregime with the idea that the dominant scattering is correctly de-scribed by a harmonic Hamiltonian, which is, in principle, trans-formable into a one-body problem of decoupled oscillators. Thethermal conductivity can then be exactly calculated by an analogof the Kubo–Greenwood formula for electrical conductivity of dis-orderedmetalswhere Anderson localization is correctly contained.In the intermediate regimebetween crystals and glasses, themixednature of propagating and localized modes lends itself to the com-plexity where both the phonon propagating and hopping descrip-tions may both fail, and characters of atomic vibrations in glassescan be classified into propagons, diffusons and locons as the notionof ‘‘phonon’’ cannot be defined [39]. Insights into the heat conduc-tion process in materials with intermediate level of disorder arelimited due to this complexity, and rely much on microscopic ex-perimental characterization and atomistic computer simulationsat a certain level of disorder.

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

The roles of disorder and heat flux localization in definingthermal 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 re-markable advances recently. Experimental evidences have demon-strated the existence of crystalline, amorphous regions and theirinterfaces in silica bilayers and graphene. Even their growth andshrinkage could be tuned and tracked under control [45–49]. These2Dmaterials thus provide an ideal platform to explore the effect ofdisorder in thermal conduction [50,51]. To gain insights into thethermal transport in 2D materials with varying levels of disorder,we performed MD simulations to analyze the results through the-oretical models of heat transport in crystals and glasses [52]. Wefound that as the level of disorder α increases, the temperaturedependence of thermal conductivity is turned over, from the sig-nature behavior of crystal to that of the glass. The critical level ofdisorder is shown to correspond to the dominance of thermal dif-fusion from local mode hops over long-range phonon propagation,and the occurrence of heat flux localization.

Specifically, MD simulation results (Fig. 3) show that thermaltransport in 2D silica with Stone–Wales type of disorder becomesdominated by the thermal hopping between localized modes withα > αcr = 0.3, which explains a turnover observed in the temper-ature dependence of the thermal conductivity. The thermal con-ductivity decreases with temperature at α < αcr, and increasesat α > αcr, and the T -dependence is weakened as α approachesαcr for both cases. The determination of the critical level of disor-der is validated from both the analysis of T -dependence turnover,dominance of diffusive contribution of thermal conductivity, the

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Fig. 3. Mechanistic transition from phonon propagation to thermal hopping as the level of disorder, α, increases in 2D silica bilayer. (a) The temperature dependence ofthermal conductivity shows a turn over at α ∼ 0.3. (b) Localization of heat flux, from non-equilibrium molecular dynamics simulations [52].

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

There are a few theoretical predictable models for heat trans-port that workwell for the limiting cases of crystals and glasses, byintroducing the microscopic mechanisms of propagating phononsor thermal hopping between localized modes. However, theapplicability of these two classes of models need to be justified formaterials with intermediate level of disorder, where either the po-larization and group velocity of localized modes or the hoppingand diffusivity of extended propagating modes are not well de-fined [54], but still make contributions to the heat conduction. Toexplore the detailedmicroscale dynamics related to this process ofthermal energy transfer, the recently isolated 2Dmaterials includ-ing a wide spectrum of crystalline lattices and types of disorder ordefects provide an ideal test-bed.

4. Interfacial thermal transport

In conventional models of interfacial thermal transport, theinterface is usually considered as a plane between two semi-infinite bulk matters and the information of interface itself isabsent [55]. Consequently, the heat carriers propagating acrossthe interface are assumed to have the same characters as those inthe bulk, in order to predict the interfacial thermal conductance.The acoustic mismatch model (AMM) presumes that all phononsare governed by continuum acoustics and there is no scatteringoccurring at the interface. The transmission probability of phononenergy is calculated from the acoustic impedance. In the oppositeextreme with strong diffusion at the interface, the diffusivemismatch model (DMM) replaces the complete secularity in AMMwith the assumption that all of the phonons are diffusely scattered.The transmission is determined by a mismatch between thephonon densities of states. As reviewed in Ref. [55], the AMMand DMM set the upper and lower bounds of interfacial thermalconductance (ITC).

With the disregard of interfacial properties, these models couldbe problematic if the vibrational coupling across the interface ismanifested by processes that are different from those contributingto the heat conduction in the materials. An apparent illustration isthat the transmission coefficient between two weakly interactedidentical nanostructures will be incorrectly predicted to be unit.Recent advances in exploring nanoscale interfaces have revealedcritical roles of the interfacial structures and interactions indefining the heat transport process across them. A number ofexperiment evidences reported in the literature demonstrate a linkbetween interfacial bonding characters and thermal conductance

at hard/soft and metal/dielectric interfaces [56–58]. This can alsobe clearly identified from the correlation between ITC for a widespectrum of nanoscale interfaces and the interfacial energy, whichare summarized in Fig. 4. The strong scattering at these weaklyinteracted interfaces permits the use of thermal resistance modelsin the interpretation of experimental or simulation resultswith theITC [59–61].

The presence of interface for a material to others may lead toleaking of propagating phonons across it and thus reduces theheat flux in it. For example, strong interface scattering of flexuralmodes lowers the in-plane thermal conductivity of supportedgraphene or graphene multilayers in comparison to suspendedgraphene [75,76]. However, this reduction may compete withother resistive mechanisms such as the scattering from surfaces.A recent work shows that the thermal conductivity of a bundleof boron nanoribbons can be significantly higher than that ofthe freestanding one [77]. Moreover, the thermal conductivityof the bundle can be switched between the enhanced valuesand that of a single nanoribbon by wetting the van der Waalsinterface between the nanoribbons with various solutions [77].For rough surfaces, the presence of free surface and nanoscaleconstriction is also critical for the heat transport. The pressuredependence of thermal conductance across contacts of roughsurfaces was reported [78]. Their results were analyzed througha quantized phonon transmission model, where the pressure-dependent conductance per atom–atom contact is Gatom = GQNτ .Here GQ = πk2BT/6h̄ is the universal quantum conductance thatsets the upper limit for flow of heat and information across a singletransport channel [8], N is the number of phonon modes and τ isthe transmission coefficients.

In nanoscale device applications where nanostructures are in-terfaced with other materials, efficient heat dissipation is neces-sary to maintain the performance and stability of nanoelectronicdevices [61,79]. Due to the limited size of contact, the electrodescontribute insignificantly as heat sinks, while at least 84% of theelectrical power supplied to the CNT electronics is dissipated di-rectly into the substrate [80]. A recent study shows that interfacialintercalation could insulate the electronic coupling at the inter-face, while the ITC is not significantly reduced [81]. As a result, anelectrically insulating but thermally transparent interfaces couldbe designed. Moreover, for these non-bonding interfaces betweennanostructures such as CNTs, graphene, and othermaterials, trans-verse phonon modes are primarily responsible for thermal cou-pling, which are different from the heat carriers in the nanostruc-tures [82,83]. For example, MD simulation results show that fora CNT on a SiO2 substrate, although inelastic scattering betweenthe CNT and substrate phonons contributes to the ITC, the ther-mal coupling is dominated by long-wavelength phonons between0 and 10 THz. The energy transfer rate is much higher for the tra-verse optical (TO)modes than for the longitudinal acoustic (LA) andtwist (TW)modes, while high-frequency (40–57 THz) CNT phonon

118 Z. Xu / Theoretical and Applied Mechanics Letters 6 (2016) 113–121

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].

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 ofheat conduction in the bulk and thus awaits an updated model forthe prediction of ITC.

5. Quantummechanical effects

In the classical treatment of atomic or ionic dynamics, allvibrational modes have the same energy of kBT according to theequipartition theorem. The quantum nuclear effect (QNE) thatis critical at low temperature or for light elements is excluded.The Debye frequency ωD defines a scale of temperature TD =

hωD/kB below which the QNE becomes prominent. One can alsoroughly estimate the characteristic temperature TQ ∼ h̄ω forthe rise of QNE significance at T < TQ, where ω is frequency

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

In exploring thermal transport, the most significant quantumeffects include the quantum statistics and the account of zero pointenergy (ZPE). One simple solution is to correct, or renormalizethe specific heat predicted from classical MD simulations. How-ever, it was pointed out that applying quantum corrections to clas-sical predictions, with or without the ZPE, does not bring theminto better agreement with the quantum predictions compared tothe uncorrected classical values for crystalline silicon above tem-

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peratures of 200 K. Mode-dependent phonon properties demon-strate that the thermal conductivity cannot be corrected on asystem level for the QNE [86]. On the other hand, a semi-quantummechanical treatment can be made, for example, through cou-pling 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 andhigh temperature limits. Recently there are a number of effortsmade to capture the quantum effects in low-dimensional thermaltransport through steady-state non-equilibrium MD simulations[87–90], which show that at temperature well below the Debyetemperature, e.g. T < 500 K for CNTs, the quantum correction toMD simulations is significant and classical results strongly overes-timate the specific heat and thus thermal conductivity. However,coupling to a quantum heat bath does not solve the problem as theZPE is represented by tangible vibrations [87].

The quantum statistical effect is included in models such as thephBPE and non-equilibriumGreen’s function (NEGF). However, thedistribution function based phBPE approach lacks of microscopicdynamics information and requires expensive computations forthe phonon frequencies and lifetimes, and the NEGF formalismis only able to calculate ballistic transport coefficients at lowtemperature where the lattice vibrations are not significant. Withthese facts, direct simulations of quantum-mechanical thermalconduction inmaterials still remain as an open challenge, althoughdirectly solving the Schrödinger equation [91], or adopting a path-integral formalism for many-particle systems [92,93], may be ableto include the QNE in a natural and feasible way.

6. Perspectives

With recent interests in developing integrated nanosystemsand nanocomposites, the defects, surfaces, interfaces have beenexplored as routes to modulate thermal transport, dissipationand management in materials. Nevertheless, there are a numberof critical issues that remain as open challenges. Phonons playa dominating role in thermal energy transport in dielectricsand intrinsic semi-conductors, as we discussed above. However,electrons and holes also act as heat carriers in doped semi-conductors and metals. Based on first-principles calculationsand the Boltzmann–Peierls equation (BPE) for phonons andelectrons, a recent work reported mode-dependent phonontransport properties of metals, where both phonon–phonon andelectron–phonon interactions are considered. The results matchwell with experimental data for non-magnetic metals and it isfound that the contribution of phonons in thermal conductionincreases in thin metal films [94]. Another calculation shows thatthe in-plane electronic thermal conductivity of doped graphene is∼300 W/mK at room temperature, independently of doping. Thisresult is much larger than expected and comparable to the totalthermal conductivity of typical metals, contributing ∼10% to thetotal thermal conductivity of bulk graphene [95]. Empirically, atwo-temperature model was constructed to include the electrondegree of freedom for thermal conduction acrossmetal–non-metalinterfaces [96].

To characterize the electronic contribution to thermal conduc-tivity, the Wiedemann–Franz (W–F) law states that the ratio be-tween the electronic thermal conductivity κe and the electricalconductivityσ at a given temperature T is a constant, i.e. the Lorenznumber L = κe/(σT ) = (π2/3)(kB/e)2, which reflects the fact thatthermal and electrical currents are carried by the same Fermionicquasiparticles [97]. TheW–F lawwas firstly proposed for bulk ma-terials and further extended to the interfacial conductance [98].For the 2D crystal graphene, this law is broadly satisfied at lowand 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 pic-ture of quasi-particle break down, it was found to be violated [99].A direct simulation of the energy transport process accountingfor both lattice vibrations and electronic transport is thus needed,which could assess validity of the W–F law and predict, for exam-ple, thermoelectric transport processes and power dissipation innanoelectronics [100].

In nanoelectronics that represent the ultimate limit to theminiaturization of electrical circuits, power dissipation remainspoorly understood owing to experimental challenges just as theheat transport. Electrical currents in the device induces over-population of phonons, which may in turn impede transportand enhance scattering with electrons as a result of strong elec-tron–phonon interaction. Joule heating and heat dissipation arethe fundamental processes for charge transport, which arise fromthe interaction between electronic charge carriers and molecularvibrations and pose serious stability issues in these devices. Re-cent works probed the local non-equilibrium between electronicand phononic temperatures in understanding the heat dissipa-tion [101]. The heat dissipation in atomic-scale junctions is foundto be intimately related to the transmission characteristics of thejunctions as predicted by the Landauer theory [60]. A correlatedelectrical–thermal transport model was developed for graphene-on-insulator devices by introducing a coupled solution of thecontinuity, thermal and electrostatic equations, offering insightsinto the formation of hot spot in device, the carrier distributions,fields and power dissipation in the graphene field effect transis-tor [102]. In addition to conduction, direct energy transfer fromhotconduction electrons to the substrate polar surface phonons (fieldcoupling) needs also to be included, especially at elevated temper-atures [103,104], and the nanoscale control of phonon excitationmay further enrich this process [105].

There have been significant and rising interest in develop-ing nano-devices for biomedical applications involving inter-faces between biological tissues and functional nanostructures[106–111]. These nano-devices are able to probe biophysiochem-ical signals, collect valuable information from key biological pro-cesses, and modulate cellular activities [106–111], which need tobe adjacent or even embedded in biological tissues. Nano-deviceswith high complexity in integration requires integrated powersupply and thus generates heat localization at the extremely smalllength scale [112]. The nano-devices made of synthetic materialsmay have material properties of high contrast to biological tissues,which thus introduce a thermal barrier to apply heat cues to reg-ulate living systems. The uncertainty about thermal managementat the interface between tissues and nano-devices has resulted ina lot of concerns and the interfacial thermal energy transfer is nodoubt pivotal in designing relevant nano-devices. Intentional ther-mal management in biological tissues has also been demonstratedas an effective control for gene express, tumor metabolism andcell-selective treatment for diseases [113–116]. Key questions thatneed to be addressed for related biomedical applications are howenergy inputs lead to local temperature rise at such a nanoscale in-terface and what criterion is to activate the thermoregulation forliving systems [117]. These concerns could be addressed by explor-ing the extreme interfaces between low-dimensional functionaldevices and cellular entities, by in vivo, in vitro, or in silico studies.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11222217) and the State Key Laboratoryof Mechanics and Control of Mechanical Structures, NanjingUniversity of Aeronautics and Astronautics (MCMS-0414G01). Thecomputation was performed on the Explorer 100 cluster systemat the Tsinghua National Laboratory for Information Science andTechnology.

120 Z. Xu / Theoretical and Applied Mechanics Letters 6 (2016) 113–121

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