Tuning Phonon Transport: From Interfaces to Nanostructures

Pamela M. Norris1

e-mail: [email protected]

Nam Q. Le

Christopher H. Baker

Department of Mechanical and

Aerospace Engineering,

University of Virginia,

122 Engineer’s Way,

Charlottesville, VA 22904-4746

Tuning Phonon Transport: FromInterfaces to NanostructuresA wide range of modern technological devices utilize materials structured at the nano-scale to improve performance. The efficiencies of many of these devices depend on theirthermal transport properties; whether a high or low conductivity is desirable, controlover thermal transport is crucial to the continued development of device performance.Here we review recent experimental, computational, and theoretical studies that havehighlighted potential methods for controlling phonon-mediated heat transfer. We discussthose parameters that affect thermal boundary conductance, such as interface morphol-ogy and material composition, as well as the emergent effects due to several interfaces inclose proximity, as in a multilayered structure or superlattice. Furthermore, we explorefuture research directions as well as some of the challenges related to improving devicethermal performance through the implementation of phonon engineering techniques.[DOI: 10.1115/1.4023584]

Keywords: nanoscale heat transfer, phonons, interfaces, superlattices, polyjunctions

1 Introduction

Thermal performance has become a limiting factor in the pro-gress of many technologies, especially with the proliferation ofdevices that incorporate nanoscale structures. Microelectronicsand energy conversion, industries of major modern relevance, fea-ture several examples of such technologies. Thermal bottleneckshave been a primary impediment to improving microprocessorspeeds [1]. The development of viable thermoelectric devices,which could scavenge energy from waste heat, depends partly onthe ability to engineer materials with lower thermal conductivity[2]. Quantum cascade lasers, a promising technology for achiev-ing terahertz frequencies in sensing and communications, remainconstrained to cryogenic operating temperatures because ofelectron–phonon interactions [3]. Phase change memory requiressimultaneous engineering of thermal and electrical properties toaccelerate switching speeds and reduce programming current,potentially leading to memory that is both faster and more energyefficient [4]. These are among the technological problems thathave motivated a great amount of research in nanoscale thermaltransport over the last few decades. In particular, structures appearin all of these applications that incorporate nano- and microscalefeatures. The thermal limitations facing these applications do nothave straightforward solutions, due in large part to relativelylimited control of thermal transport compared to electronic trans-port [5].

Hence, in this paper we survey recent research on phonontransport in layered structures, especially in the context of heattransfer. Models of phonon transport in bulk materials are rela-tively well developed, but in nanoscale systems, interfaces andboundaries are at least as important as bulk properties to overalltransport. This is true of structures in all of the opening examples.Despite decades of research, models of interfacial effects remainmuch less mature, and are a major impediment to understandingtransport in layered structures. A detailed background discussionof phonon transport physics is beyond the scope of this review,and is covered in texts such as the one by Chen [6]. We also referto the previous review by Cahill et al. [7] for extensive discussionof phononic heat transfer in nanoscale systems to that point,and to the recent review by Pop [1] especially in the context of

microelectronics. The effects of interfaces on transport can bebroadly categorized by comparing characteristic layer thicknessesd with the mean free paths k of phonons that contribute signifi-cantly to transport. This distinction can be simplistic because ofthe broadband nature of phonon transport and the potentiallystrong variation of k for different phonon frequencies. However,in the far limit that d � k, a layered structure is effectivelydecoupled into a series of single interfaces. We discuss singleinterfaces in Sec. 2, with emphasis on the ability to tune their ther-mal conductance. Many interesting structures also fall in regimeswhere d � k or d � k. In these cases the analysis of phonon scat-tering at a single interface is affected by the proximity of nearbyinterfaces. For the purposes of our discussion, we divide these lay-ered structures into two broad categories, again with attention tostrategies for tuning thermal transport. In Sec. 3 we discuss super-lattices and multilayers; that is, extended structures with manylayers, typically repeating between two materials. In Sec. 4 wediscuss polyjunctions, which we define as structures with fewlayers in which repetition is not an important feature. In each sec-tion we also discuss potential future research problems relevant tothose structures. Finally, we summarize the review in Sec. 5.

2 Interfaces

Interfaces have an intrinsic resistance to heat flow across themdue to phonon scattering caused by abrupt changes in crystallogra-phy. In order to facilitate device thermal modeling, the micro-scopic details of interfacial phonon transport are aggregated into asingle macroscopic parameter, the thermal boundary conductancehBD, which is the inverse of the thermal boundary resistance, thatrelates the temperature discontinuity at the interface to the appliedflux:

q ¼ hBDDT (1)

where q is the heat flux normal to the interface, and DT is the tem-perature discontinuity. Since the observation of hBD in 1941 byKapitza [8], researchers have sought a predictive model for hBD.At the most fundamental level, phononic properties depend on thewavevector and phonon transport is described in wavenumberspace. However, it is convenient to perform analysis in frequencyspace, in terms of both ease of calculation and interpretation. For

1Corresponding author.Manuscript received October 17, 2012; final manuscript received December 23,

2012; published online May 16, 2013. Assoc. Editor: Leslie Phinney.

Journal of Heat Transfer JUNE 2013, Vol. 135 / 061604-1Copyright VC 2013 by ASME

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an ensemble of phonons incident on an interface, the transportacross is described within the Landauer formalism by [6]

hBD ¼1

2

ð1

0

Xj

ðxc;j

0

�hxvjðxÞaj;1!2ðx;lÞDjðxÞ@f ðx;TÞ@T

@x

" #ldl

(2)

where j denotes a phonon branch, xc;j is the cut off frequency forbranch j, �h is the reduced Planck constant, x is the angular phononfrequency, vj is the phonon group velocity, aj;1!2 is the transmis-sion probability from side 1 to side 2, l is the cosine of the inci-dence angle, Dj is the spectral phonon density of states of branchj, and f is the Bose–Einstein distribution. This formulation isderived from the wavevector description under the assumptions ofsmall temperature differential across the interface and isotropicdispersion. The main difficulty in carrying out this calculation isin the prediction of the transmission probability, which differenti-ates the various approaches to modeling hBD.

Early attempts at modeling the thermal boundary conductanceused an analogy to classical wave theory and acoustics. Theresulting acoustic mismatch model (AMM) calculates the trans-missivity based on the acoustic impedances of the constituentmaterials, giving in the simplest case of a phonon with transversepolarization perpendicular to the plane of incidence [6,9]:

aj;1!2ðlÞ ¼ 1� Z1l� Z2cosht

Z1lþ Z2cosht

� �2

(3)

where Z is the acoustic impedance of the material given by theproduct of Brillouin zone center group velocity and densityZ ¼ qvjð0Þ, where ht is the angle of the transmitted phonon com-puted by Snell’s law, and only the acoustic branches are consid-ered. In using the zone center group velocity (sound speed) foreach phonon, the model implicitly assumes a Debye type disper-sion. The Debye dispersion is accurate in the limit of low wave-numbers; thus the acoustic mismatch is applicable to interfaceconductance at very low temperatures, when the low wavenumberphonons are the dominant energy carriers. The model alsoassumes completely specular scattering, which is applicable to theextent that the phonon wavelength is greater than the scale ofinterface roughness. Thus, the accuracy of the AMM increases forsmoother interfaces at low temperatures.

Yet, most actual interfaces display some measure of roughnessor other imperfections and the regime of interest is above cryo-genic temperatures, leading to a breakdown of the assumptionsthat went into the AMM. The diffuse mismatch model (DMM)was derived for these types of circumstances and gives the trans-missivity as [10]

a1!2ðxÞ ¼

Xj

v2;jD2;jXj

v2;jD2;j þX

j

v1;jD1;j

(4)

where the summation across phonon modes means that thephonons can scatter into any available branch. This formulationmodels phonon transport across the interface as completely diffu-sive, i.e., the angle of the transmitted or reflected phonon and itspolarization bear no relation to the incident phonon. In this case,the transmissivity is weighted by the group velocity and density ofstates of the two materials. The DMM succeeds in capturingaspects of nonideal interfaces for which the diffuse scatteringassumption is valid. The AMM and DMM represent a reasonablelower limit and upper limit, respectively, to hBD when consideringelastic phonon scattering,2 where phonon frequency is conservedduring the scattering event.

Both the AMM and DMM capture the aspects most importantto interfacial thermal transport: the mismatch in materialvibrational properties and the phonon population incident on theinterface. The AMM takes the acoustic impedance as the criticalvibrational property, while the DMM considers the product ofgroup velocity and the density of states. The phonon population isdictated by the temperature, which enters the model through theBose–Einstein distribution and the density of states. Nevertheless,the treatment thus far of hBD has neglected the characteristics ofthe interface itself. Several modifications have been made to theDMM in recent years in order to more accurately calculate thehBD of interfaces where the structure is known to diverge fromthe ideal interface. The use of Debye dispersion has been replacedby isotropic dispersion with [12] and without [13] optical branchcontributions or full dispersion [14], inelastic scattering has beenconsidered in addition to elastic scattering [15–17], and the effectsof interface disorder [18–22] have all been recently considered.

The myriad issues that have been addressed in order to makethe DMM conform to experimental observations illustrate thecomplexity of the problem of interface conductance while provid-ing an overview of the factors that determine it. The two factorsthat have the greatest influence on hBD are the vibrational mis-match of the materials that comprise the interface and the temper-ature of the system. The former is included explicitly in both theAMM and the DMM, and the temperature dependence manifestsitself not only in the change in the phonon population, but alsothrough a change to the phonon dispersion and increased inelasticscattering at the interface [17]. The interface bond strength andsystem pressure also affect hBD by modifying the mode dependenttransmissivity of different phonons. The mismatch, temperature,interface bond strength, and pressure are all parameters which,although they influence hBD, are characterized by limited controll-ability in actual device applications. However, modifications tothe local interface structure offer the ability to tune hBD independ-ently of bulk material and environmental conditions. Parametersthat belong to this category include the interface roughness, inter-diffusion, and defect density. The ranges over which hBD havebeen experimentally tuned using these methods are collated inFig. 1.

2.1 Interface Vibrational Mismatch. The relative vibra-tional properties of the materials that constitute the interface

Fig. 1 Tuning of hBD achieved in experiments by roughness(Sec. 2.5), interdiffusion (Sec. 2.6), and defects (Sec. 2.7) atroom temperature. In actual systems, each interface conditionwill be accompanied, to some extent, by the others. Thoroughcharacterization of the interface is essential in experimentsseeking to understand the effects of interface conditions onhBD .

2The phonon radiation limit is a model that assumes a transmissivity of one for allincident phonons [11], although discussion in this work is limited to theaforementioned mismatch models and their derivatives.

061604-2 / Vol. 135, JUNE 2013 Transactions of the ASME


figure prominently in both the AMM and DMM. In general,materials with similar vibrational properties will have a highertransmissivity, while the converse is true for mismatched materi-als. Using the Debye temperature as a proxy for the vibrationalproperties of materials, Stoner and Maris [11] confirmed that hBD

tends to decrease with an increase in the mismatch. At largerDebye temperature mismatches, hBD exceeded the maximumvalue permitted by elastic scattering, meaning that inelastic scat-tering channels have a significant effect. This result is supportedby Stevens et al. [23] and Lyeo and Cahill [24], who found thathBD of highly mismatched interfaces exceeds that predicted by theDMM and phonon radiation limit, respectively.

Molecular dynamics simulation offers the ability to isolate theeffects of changes to the vibrational mismatch between materi-als. Simulations have been conducted that vary the vibrationalspectrum of one side of the interface, often by altering the mass,but also by adjusting the interatomic potential. Twu and Ho [25]used the potential well widths of the Lennard-Jones and Morsepotential as a proxy for the mismatch and found an approxi-mately exponential decrease in hBD with increased mismatch.Using the ratio of Debye temperatures, Stevens et al. [26] founda decrease in hBD with greater mismatch. Hu et al. [27] adjustedthe bond strength of silicon in a Si/polymer system. They founda power law decrease in hBD with an increase in the stiffness ofthe silicon. Landry and McGaughey [28] modified the mass ofsilicon in a Si/heavy-Si system and found that hBD decreasedwith increased mismatch. Lyver and Blaisten-Barojas [29] con-firmed the trend of decreased hBD with mismatch. Wang andLiang [30] examined a Ar/heavy-Ar interface and found a linearincrease in thermal boundary resistance with mismatch. Thetrend was also confirmed by Ju et al. [31]; they show that abovea mass ratio of 3, the boundary conductance as computed in MDis less than the DMM and AMM predict, whereas it is higherbelow 3. Finally, Shen et al. [32] found a linear trend of con-ductance with mass ratio.

The issue of vibrational mismatch is further complicated bythe fact that the density of states is altered from that of the bulknear the interface [29,31,33–35]. Specifically, high frequencymodes appear and the lower frequency modes are softened in thesofter material [29,33]. The effect of the interface on the localdensity of states was not observed in the work of Ju et al. [31],although, according to the authors, this could be because thesmall system sizes studied diminished the distinction betweeninterface and bulk atoms. The alteration of the density of statesin the vicinity of the interface and the effect of this on hBD isdifficult to ascertain. Experimental observation of this effect, andincorporation of it into future models of hBD, may improve theability to predict hBD. It is likely that the local density of stateswill affect incoming phonons differently based on their wave-length and/or mean free path in comparison to the scale of theaffected interfacial region.

2.2 Interface Temperature. In the elastic formulation ofthe DMM, hBD has temperature dependence only through theBose–Einstein distribution. Consequently, the lower Debye tem-perature among the interface materials is a critical temperaturewhen considering transport. Below this temperature, hBD increaseswith temperature as more modes become available to participatein the transport. In the low temperature limit, hBD grows as T3

according to the Debye model of heat capacity [6]. Above thelower of the two Debye temperatures, Eq. (2) predicts a constanthBD since there are no new modes available for elastic scattering,either on one side of the interface or the other. However, experi-mental studies of the temperature trend by Lyeo and Cahill [24]and Hopkins et al. [36,37] show a clear increase in hBD withtemperature, in disagreement with theories discussed so far. Thisdisagreement is attributed to the onset of inelastic phonon scatter-ing, whereby multiple phonons can combine and/or split whenscattering at the interface, relaxing the restrictions imposed byconservation of energy.

Once the system temperature exceeds the larger Debye temper-ature, the system has reached the classical regime where all pho-non modes are excited. These conditions correspond to thermalmodeling of materials with classical molecular dynamics simula-tion, where all vibrational modes are excited at all temperatures.Molecular dynamics simulations by Stevens et al. [26], Hu et al.[27], Luo and Lloyd [38,39], and Duda et al. [34] show a linearincrease in hBD with temperature. Other simulations also support amonotonically increasing thermal boundary conductance withtemperature [28,30,31,40]. This result supports the experimentalfindings. Furthermore, the same mechanism proposed to explainexperimental results is also present in the simulations; anharmo-nicity of the interatomic potential permits inelastic scattering.

Inelastic phonon scattering at the interface has been treatedtheoretically by Hopkins et al. [16,17] and Duda et al. [41] usingtwo different mechanisms. In the work of Hopkins et al. [16,17],the two-phonon elastic transmission is calculated normally andthen the principle of detailed balance is applied to the remainingphonons that would have been reflected had they been unableto participate in inelastic processes. This is repeated for up to N-phonon processes, resulting in better agreement with experimentaldata. In the second mechanism, Duda et al. [41] proposed that thehigh temperature linear trend is actually due to phonon-phononscattering. In their model, it is argued that low wavenumber pho-nons can ignore the interface and undergo phonon-phonon scatter-ing after traversing the interface. This process is reminiscent ofevanescent modes penetrating into a material [6]. The temperaturedependence arises from models of thermal conductivity where thephonon-phonon scattering rate is proportional to the temperatureat high temperatures [42–44].

2.3 Interface Bonding. The fact that bulk material propertiesare used in the formulation of the AMM and DMM implies thatthey model an ideally bonded interface, that is, one with an infi-nite spring constant [45]. In actual materials, the bond strength atthe interface is finite and its value affects hBD. Young and Maris[46] used a lattice dynamical model to investigate the effects ofthe interface bonding in a spring-mass system. They found thatthe bonding had little effect for the range of strengths betweenthose of the bulk materials. Above this range, the effect of bondstrength was slight, and for bonds at the interface weaker thanthose in the bulk, hBD scaled quadratically with the bond strength.The quadratic trend for weak interfaces has been confirmed insubsequent, acoustical modeling [45,47]. In these models, two dif-ferent parameters are used to characterize the interface bondingcondition: the bond energy and the spring constant j, and bothshow the same trend.

To elucidate the relationship between material interfaces, bondenergy, and spring constant, we consider the Lennard-Jonespotential. Although this potential is only well suited for themodeling of van der Waals interactions, it captures the basicunderlying principles common to all types of bonding. The formof the potential is

u ¼ 4er12

r12� r6

r6

� �(5)

where u is the energy of the pair interaction, e gives the depth ofthe energy well, r is a length parameter determining equilibriumseparation distance, and r is the separation distance between twoatoms. Taking a Taylor expansion of the potential about the equi-librium separation distance, it can be seen that j / e, whichexplains why hBD has the same trend with respect to both thespring constant and bond strength in the models. Most real interfa-ces will be in the regime of growing hBD with stronger bondingbecause of considerations of interface thermodynamic stability.One condition for a stable interface is that the energy associatedwith those bonds must be higher (less negative) than the bonds inthe bulk [48], otherwise the materials will mix in order to obtain a

Journal of Heat Transfer JUNE 2013, Vol. 135 / 061604-3


lower system energy.3 This places the typical interface in theweakly bonded regime described by Young and Maris [46].

Molecular dynamics simulations have confirmed the generalrelationship between bond strength and hBD, although simulationshave shown a roughly linear trend for Si/amorphous-polyethylene[27], carbon nanotube/SiO2 [50], Si/Ge [51], Cu/SiO2 [52], andLennard-Jones solid-solid [32] and solid-liquid [53] systems. Thereason for the difference between the quadratic trend predicted bytheories concerned with interfacial bonding [45,46,47] and theobserved trend in simulations is unclear.

Experimentally, Losego et al. [54] harnessed the effect ofbonding to tune hBD at a Au/self-assembled-monolayer (SAM)junction by varying the functional group termination of theSAM. Using this method, the authors were able to tune hBD overthe range of about 34 to 65 MW m�2K�1. Also using a molecularlayer approach, O’Brien et al. [52] tuned the overall hBD of aCu/molecule/SiO2 interface over the range of 260 to430 MW m�2K�1, which also represented a dramatic increaseover the baseline Cu/SiO2 hBD of 90 MW m�2K�1. Collins et al.[55] tuned hBD between aluminum and single crystalline diamondusing hydrogen and oxygen surface treatments of the diamond,achieving tuning in the range of 19 to 91 MW m�2K�1. The oxy-gen terminated diamond surface offered stronger bonding with thealuminum than the hydrogen, leading to higher values of hBD.Using a similar method, Hopkins et al. [56] functionalizedsingle layer graphene with oxygen or hydrogen to tune the hBD

of an Al/graphene/SiO2 interface in the range of 23 to42 MW m�2K�1. The use of SAMs and adatoms to improve hBD

parallels current research into inorganic materials for the samepurpose (see Sec. 4.1). As this approach becomes more common,these results indicate that chemical functionalization will be aneffective method for tuning hBD.

2.4 Interface Pressure. Related to the effect of bondstrength is the effect of applied pressure. The application ofpressure alters the equilibrium separation distance of atoms, dis-proportionately affecting atoms with weaker bonding. Thischanges both the vibrational properties of the materials compris-ing the interface and the force interaction between interfacialatoms. By changing the vibrational properties of the bulk materi-als, the mismatch is changed, altering hBD as discussed inSec. 2.1. Simulations by Shen et al. [32] and Liu et al. [57], andexperiments by Hsieh et al. [58] indicate that the interfacial ther-mal conductance increases linearly with the applied pressure forweakly bonded interfaces. Referring to Eq. (5), by expanding thepair potential in a Taylor series, it can be shown that a compres-sive pressure increases both the harmonic and higher order forceconstants and vice versa for a tensile pressure. Thus, the pressuretunes hBD primarily through the change to the interface springconstant, but also through a change to inelastic scattering.

2.5 Interface Roughness. We now turn to those methodsthat offer the ability to tune hBD independently of the interfacematerials and environmental conditions. A roughened interface isone for which the interface is nonplanar, yet still exhibits a sharpchange in material composition. There are a couple of parametersto quantify the roughness of an interface. The first is the rootmean square (rms) height of the interface features. The second isthe correlation length, which is a measure of the lateral distancebetween similar features of the interface topology [59]. Bothparameters can be derived from measurements of the surface pro-file before deposition of the second material, using a microscopytechnique such as atomic force or scanning tunneling microscopy.

Theory predicts that roughness will increase hBD due to anincrease in diffuse scattering. Kechrakos [60,61] predicted an

increase in hBD across all temperatures for an interface withroughness on the order of a monolayer and an enhancement ofabout 2 to 3 times for a roughened interface above the Debye tem-perature for moderately mismatched materials. Fagas et al. [62]also theoretically treated interface roughness in the limit of a sin-gle monolayer. Varying the correlation length of mass impuritiesat the monolayer thick interface, they found that the phonon trans-missivity has a strong dependence on phonon frequency and theroughness correlation length. Both rms roughness and correlationlength can be incorporated into a single Fourier spectrum descrip-tion. Taking this approach with a Green’s function analysis, Zhaoand Freund [59] showed that the specularity of the interfacedepended primarily on the interface rms roughness and not thecorrelation length. Even so, in the limit of a small interface tem-perature discontinuity, the authors found that hBD was relativelyinsensitive to both the rms roughness and correlation length, atleast over the range of values investigated. A study by Sun andPipe [63] used a modeling approach that treats roughness as a per-turbation to a perfectly flat interface. Their results largely corrobo-rate those of Zhao and Freund [59], in that the correlation lengthof the roughness does not have a strong effect. Their work coversmany of the aspects of specular and diffuse scattering. Roughnessacts to decrease specular transmission and reflection for eachpolarization across all incidence angles. In comparison to theDMM, they found that transmitted (and reflected) modes are pref-erentially concentrated near the incident phonon wavenumber.Only the amplitude of this distribution is affected by the mismatchin material vibrational properties. Finally, the authors demonstratethe presence of evanescent modes under certain conditions of pho-non incidence.

In contrast to the theoretical results that predicted an increase inhBD with roughness, experimental observations have indicated astrong, decreasing trend. Decreasing hBD with roughness hasbeen observed experimentally for an Al/Si interface [20,64], Al/GaAs interface [21], and an Al/Si interface with quantum dotroughening [22]. One explanation for the opposite trend is that,experimentally, a roughened interface is typically accompaniedby various crystallographic defects, obfuscating the direct rela-tionship between the roughness and hBD. Nevertheless, theauthors were able to use roughness to tune hBD in the range ofabout 90 to 193 MW m�2K�1 for the Al/Si interface [64], 15to 29 MW m�2K�1 for the Al/GaAs interface [21], and 105 to170 MW m�2K�1 for the quantum dot interface [22]. In theseexamples, interface roughness was achieved through wet chemicaletching, deposition conditions, and quantum dot growth condi-tions, respectively.

2.6 Interface Interdiffusion. Interdiffusion of the twomaterials comprising the interface is another factor that influenceshBD. Whereas interface topology is the defining parameter ofroughness, the species concentration distribution across the inter-face is the key parameter for interdiffusion. Transport across dif-fuse interfaces was treated theoretically by Beechem et al. [18],where the disordered region was treated as its own material withweighted properties of the elements involved. Their model pre-dicts a minimum in hBD for intermediate mixing of materials inthe alloyed interfacial region and shows reasonable agreementwith experimental results.

Simulations investigating the effects of interdiffusion displayvaried results. Simulations by Liang and Sun [65], Stevens et al.[26], and English et al. [35] of Ar/heavy-Ar systems show anincrease in hBD with moderate interdiffusion. This is in contrast toother simulations by Twu and Ho [25], Li and Yang [51], andChoi et al. [66] that found a decrease in hBD for an Ar/Ar system,a Si/Ge system, and an Al/Si system, respectively. A notable dif-ference between these sets of simulations is that the materialswere lattice matched when hBD increased and not lattice matchedwhen it decreased. The relationship between interdiffusion andhBD may also be sensitive to the particular functional form of the

3Interface stability in certain material systems may also arise due to slowdiffusion kinetics [49]. Nevertheless, for present purposes, thermodynamics offers asufficient explanation linking interfaces and bond strengths.



concentration profile, making comparisons between different sim-ulations (and experiments) tenuous [35].

Experimental results of a Cr/Si interface by Hopkins et al.[19,67] show that interdiffusion can be used to tune hBD. The stud-ies showed that the largest conductance values were obtainedwhen the concentration gradient of the species transitioning acrossthe interface was small. The concentration gradient was controlledthrough the deposition conditions. The thickness of the mixingregion also played a role, but perhaps only insofar as a thickermixing region enables a smoother transition from one material tothe other. Using interdiffusion, the authors tuned hBD in the rangeof about 113 to 178 MW m�2K�1. The complex role of the speciesconcentration distribution is supported by the simulation study byEnglish et al. [35], who show that the thermal boundary conduct-ance is a nonmonotonic function of the degree of interdiffusion.

2.7 Interface Defects. Just as with bulk conduction, the pres-ence of defects at the interface provides an additional mechanismby which phonons can scatter, affecting hBD. All nonidealities ofan interface that do not correspond to roughness or interdiffusion,as discussed in previous sections, will be treated under the samebroad category of defects in this section. Defects arising fromlattice mismatch and impurity atoms are two examples. The dis-tinction among the various interface nonidealities is complicatedby the fact that both roughness and interdiffusion invariably intro-duce other crystallographic defects, and that models developed fornonideal interfaces do not always attempt to distinguish betweenthe different aspects, as covered here.

Models by Kozorezov et al. [68] and Prasher and Phelan [69],while developed for a generic, nonideal interface, will beapplied here as meaning an interface with defects. The model byKozorezov et al. [68] treats a defected region in the vicinity of theinterface as its own material and determines the relative contributionsof specular and diffuse scattering. The model by Prasher and Phelan[69] predicts hBD, while considering phonon scattering in the vicinityof the interface; i.e., the phonons that interact with the interface maynot have a bulk-like distribution. Simulations by Li and Yang [51]display a reduction in hBD with vacancy size and concentration.

Experimental results also indicate a decrease in hBD with defectdensity. In one study, Hopkins et al. [21] tuned the straindislocation defect density of a GaSb/GaAs interface; modifyingthe width of the defected region leads to tuning of hBD from 12 to21 MW m�2K�1. In another study, Hopkins et al. [70] used protonion implantation to add defects to Al/Si and Al/Al2O3 interfaceswhich led to the ability to tune hBD in the range of 25 to200 MW m�2K�1 and 12 to 310 MW m�2K�1, respectively,although mechanisms beyond the defects may also have contrib-uted [71]. Norris et al. [72] used various surface cleaningmethods to alter the interface structure and chemical impurities ofa Au/highly-ordered-pyrolytic-graphite system, leading to theability to tune hBD in the range of 7 to 31 MW m�2K�1.

Although it does not offer the same control and range of tuna-bility, the deposition method and parameters offer an additionalroute to tuning hBD. In a study by Kato and Hatta [73], Au wasdeposited on SiO2 using evaporation and sputtering with differentdeposition rates. The hBD of the evaporation deposited interfacewas about 122 MW m�2K�1, and for the sputtered interface it wasabout 31 MW m�2K�1. In a similar study, Monachon et al. [74]also examined the effect of evaporation and sputtering on hBD.The surface quality before deposition affected the relationshipbetween the evaporated and sputtered values. They measured anevaporated conductance of 43 MW m�2K�1 and a sputtered valueof 171 MW m�2K�1 on the untreated interface. On an interfacecleaned by reactive-ion etching, the values were 241 and186 MW m�2K�1 for the evaporated and sputtered interface,respectively.

2.8 Interface Research Outlook. In this section we havediscussed investigations of interfacial thermal transport by theo-

retical, simulated, and experimental means, with attention to howinterface structure can be modified in order to tune phonon trans-port. Significant progress has been made in the understanding andengineering of interfacial thermal transport in the past decade. Ofparticular note are the strides made in the realms of modeling andsimulation where increased computing power has enabled virtualstudies of great complexity and sophistication, complementingthe experimental investigations. Yet, we now turn our attention to-ward specific aspects of research where we believe coordinatedeffort would greatly profit the field of nanoscale thermal transport.

First, we note that our discussion has been restricted to interfa-cial transport involving phonons only. Experimental evidence hassuggested that metal/insulator interfaces can often be treated asphonon dominated, neglecting the direct energy transfer betweenelectrons in the metal and phonons in the nonmetal [11,24]. How-ever, some theoretical [75–77] and experimental [78,79] studieshave shown that the transport through this channel might be sig-nificant, and attempts to tune interfacial conductance may need toconsider this mechanism in some systems, such as the Au/Si inter-face. Although a detailed discussion of electron-phonon interac-tions is beyond the scope of this work, we highlight this as animportant question for future research, given the ubiquity ofmetal/nonmetal interfaces in device applications, as well as theirimportance in thermal metrology at the nanoscale.

Despite the numerous modifications to the DMM that haveimproved its correlation with experimental data, many interfacesare not completely diffusive, lying somewhere between specularand diffuse on the spectrum of interface conditions [63,80].Recent simulated results that demonstrate the modification to thedensity of states near the interface, compared to the bulk, implythat, from the perspective of the phonons, the interface is a muchlarger construct than a mere infinitesimal plane [29,31,33–35],extending at least as far as the strain relaxation distance. As such,we encourage pursuit of a model that treats phonon scattering inthe bulk, near an interface, and at the interface using the samegeneral framework. In this direction, one wonders if the scatteringrates can be inferred from the variation in the occupied density ofstates near an interface.

Simulations have demonstrated their worth with respect to iso-lating parameters that influence thermal boundary conductance,measuring system properties and dynamics with atomic resolu-tion, and determining trends. However, there are, as of yet, veryfew studies that attempt to recreate experimental conditions wheretwo or more mechanisms influence thermal boundary conduct-ance. The increasing accessibility of computing power shouldallow such inquiries into the atomic-scale explanations of trans-port behaviors that are experimentally observed.

Finally, if interfacial structure is to be harnessed in order totune thermal transport in actual devices, a highly detailed under-standing of the interfacial structures that correspond to measuredvalues of thermal boundary conductance is required. We encour-age experimentalists to characterize the interfacial structures thatthey study to the fullest extent possible. This will facilitate thedevelopment of models and help to isolate the contributions ofdifferent interface mechanisms, better enable simulations to cor-roborate results, and, ultimately, hasten the technological imple-mentation of sophisticated, engineered interfaces.

3 Multilayers

Multilayers, and the special case of superlattices, are structurescomposed of many layers in a periodic sequence. They haveinspired much research related to low phononic thermal conduc-tivity k, which arises from suppressed phonon transport. Thesestructures are relevant in a variety of applications; in somesystems, the low k is a desirable feature (e.g., thermoelectrics,thermal barriers), while in others it might be a side effect of engi-neering other properties (e.g., optoelectronics such as quantumcascade lasers). In either case, the ability to engineer thermalproperties requires an understanding of underlying phonon



behavior. Besides their relevance in these applications, superlatti-ces in themselves also provide a rich context for studying the fun-damental science of phonon transport. To date, research hasfocused on structures alternating between two materials (i.e., twolayers per period, ABAB.). Regarding terminology, we will usethe term “multilayer” to refer to these structures in general. Wewill follow the convention of reserving the term “superlattice” forthe special cases when the layers are themselves single crystalline,in which case the combined symmetries of the crystal lattice andthe superlattice are potentially important. However, both termsare used with varying scope in the literature, and the distinction isnot always clear; the transport in these structures is complex anddefies easy categorization.

3.1 Models for Multilayer Transport. The comparisonbetween the thickness of layers in a multilayer (or, similarly, theperiod length L) and the dominant phonon mean free paths k oftenserves as a rough proxy for the distinction between diffusiveand ballistic phonon transport. Phonons that dominate thermaltransport in common crystalline materials are thought to have kspanning several orders of magnitude, 10 nm to 10 lm [7,81,82].When layer thicknesses are much greater than k, many phonon-phonon scattering events occur in the material between interfaces.Phonon transport is diffusive, and heat transfer is modeled well bya conventional thermal resistance network. Section 2 discusses thevarious nanoscale effects involved in analyzing interfacial resis-tances. Otherwise, we will not discuss this case here.

We focus on the regime where distances between interfaces areon the scale of k and smaller. This corresponds to a transition toballistic transport, in which interface scattering dominates and lit-tle phonon-phonon scattering occurs in the layers. The resistancenetwork model for heat transfer breaks down. Within this regime,phonon transport may also differ in exhibiting particlelike (inco-herent) and/or wavelike (coherent) behavior, as distinguished bythe presence of interference among phonons. The Boltzmanntransport equation (BTE) serves as the typical basis for incoherentmodels [83–86]. The dominant mechanisms that suppress thermalconductivity are interface and defect scattering, which areassumed to be sufficiently diffusive to destroy coherence. In theother limit, coherent transport models are typically based on har-monic lattice dynamics (LD) to describe wavelike propagation ofenergy through the atomic lattice [87–92]. This presumes thatinterface scattering is specular and preserves the phase of scat-tered phonons. In a crystalline superlattice, the phonon dispersionrelation reflects the extended periodicity L > a in the cross-planedirection (lattice parameter a). In reciprocal space, this shrinks theedge of the first Brillouin zone compared to the bulk, and the pho-non dispersion relation is folded inside. This has two importanteffects: reduced group velocities and the emergence of phononicband gaps. A smaller Brillouin zone would also mean differentcriteria for umklapp phonon-phonon scattering processes, whichRen and Dow termed “mini-umklapp” processes [93].

The relative impact of particle- and wavelike behaviors onaggregate heat transfer in real systems is the subject of somedebate, and the issue is addressed throughout this section. To com-plicate matters, phonons in real superlattices are likely to exhibitsome mixture of these behaviors. In addition, the various suppres-sion mechanisms exhibit different frequency dependence, whichis important because of the broadband nature of thermal phonons.Molecular dynamics (MD) simulations have been a useful tool formodeling the complex interplay between these effects [94–102]since they require no prior assumption of ballistic/diffusive trans-port or particle/wave behavior. The simulations allow these dis-tinctions to emerge from atomistic behaviors, making MDespecially powerful for investigating the factors responsible fortransitions between ideal cases. However, we note that MD simu-lations have been better suited for investigating general trendsthan for making quantitative predictions since they capture onlyclassical effects, and current interatomic potentials are only rough

approximations to true interactions in all but the simplestmaterials.

3.2 Experimental Measurements of Multilayers. Theaforementioned models provide predictions for different limits ofphonon behavior—for example, how coherent and incoherentbehavior might make different contributions to the thermal con-ductivity as a function of period length kðLÞ. These predictedtrends are templates for experimental validation; thus (ideally)enabling researchers to deduce the dominant mechanisms in realmultilayers. This dates back to the experiments of Narayanamurtiet al. [103], in which the generation and transmission of mono-chromatic phonons exhibited a clear stop band in GaAs/AlGaAssuperlattices, signifying coherent transport. Subsequent experi-ments have reinforced the observation of coherence in varioussuperlattices of III-V materials [104–106]. However, in the con-text of phononic heat transfer, the investigated phonon frequen-cies are low—on the order of hundreds of GHz. Although theseexperiments provide clear evidence of coherence at low frequen-cies (long wavelengths), defects and interfacial features affectshorter-wavelength phonons more strongly, potentially disruptingcoherence. Hence, the significance of coherence in aggregate,broadband thermal transport is less clear, and has been inferredonly indirectly from thermal measurements.

Measurements of thermal properties in superlattices date backto Yao [107], who used ac calorimetry to measure the in-planediffusivity and conductivity of AlAs/GaAs superlattices with dif-ferent period lengths. Yu et al. [108] used the same method togather temperature-dependent data. Thermal measurements ofmultilayers accelerated with the development of the 3x technique[109–119] and thermoreflectance techniques [120–123], whichhave been used to measure superlattice conductivity k primarily inthe cross-plane direction. In the context of thermal properties, wewill emphasize research on cross-plane transport since it hasshown greater potential for thermal engineering.

Some important trends in experimental data, especially k, moti-vate the discussion in the following subsections. The dependenceof k on temperature (Sec. 3.3) has proven most straightforward tointerpret across reports. Weak temperature dependence acrossmaterial systems implies that interface and defect scattering arethe dominant suppression mechanisms in most systems, even withsome reports showing potential coherent effects. The dependenceof k on period length (Sec. 3.4) has been a more complex subject,and overall trends in the literature are more difficult to discern.Reported trends appear strongly dependent on material systems.This suggests a large role for interfacial quality and defects, whichwe discuss in Sec. 3.5. These combined insights have set the stagefor some recent examples of multilayer structures with deliberatethermal design, which we discuss in Sec. 3.6 along with possibleavenues for future investigation.

3.3 Temperature in Multilayers. Temperature is not typi-cally a free design parameter in device applications, so we discussit more for insight into underlying transport than as a practicalmeans for controlling conductivity. Among the discussed mecha-nisms that suppress phonon transport, only umklapp scattering isstrongly temperature dependent. This creates the T�1 conductivitytrend in bulk crystals at high temperatures, where phonon-phononscattering dominates. However, for multilayers in the ballisticregime, we assume that this is much less important than interfaceand defect scattering, which are temperature independent, at leastunder the basic assumption of elastic phonon scattering. There-fore, the cross-plane conductivity in multilayers is expected onlyto increase with low temperatures due to quantum mechanicalphonon population effects; at higher temperatures, k should onlybe weakly temperature dependent.

The possible exception arises if coherent effects are significant,giving rise to mini-umklapp scattering as a significant mechanism.The early theoretical report by Ren and Dow [93] analyzed this



case, predicting a T�1 conductivity trend at high temperatures.However, they predicted that coherent effects would suppress con-ductivity only weakly in a Ge-like superlattice (<30% reduction)and warned that competing suppression from defect scatteringcould mask the coherent effect. In MD simulations, McGaugheyet al. [97] reported a strong inverse relationship between k and T,corroborating the theoretical predictions. In addition, they drew adirect connection to anticipated changes in the calculated phonondispersion: reduced group velocities and the emergence of stopbands. However, they demonstrated this effect only for idealsuperlattices.

Experimental measurements of cross-plane conductivity, com-piled in Fig. 2, have been generally consistent with theoreticalexpectations. In structures where no coherent transport isexpected, k shows no decrease with T since neither umklapp normini-umklapp processes should be important. Such trends areobserved in the layered Si/Si0:7Ge0:3 and Si0:84Ge0:16/Si0:76Ge0:24

structures studied by Huxtable et al. [115], in which alloy scatter-ing should preclude coherence in all but the lowest-frequencyphonons. The same pattern is seen in the data reported by Cost-escu et al. [122] for W and amorphous Al2O3.

Among superlattices grown epitaxially, evidence for the signifi-cance of coherent effects is mixed. As seen in Fig. 2, somedecrease in k with T is seen in the data from Capinski et al. forGaAs/AlAs [120], from Borca-Tasciuc et al. for InAs/AlSb [114],and from Cahill et al. for PbTe/PbSe [113]. However, the Si/Gestructures studied by Lee et al. [109] and Borca-Tasciuc et al.[112] showed no sign of decreasing k with T. In these cases, thelattice mismatch between Si and Ge appeared to play a large rolein disrupting coherence. The report by Song et al. [111] alsoshows no decrease in the conductivity of skutterudite IrSb3/CoSb3

superlattices. The dominant explanation for these mixed reports ofcoherence has been the disruptive effect of scattering at nonidealinterfaces, as expected by Ren and Dow [93].

3.4 Period Length in Multilayers. Unlike temperature, theperiod length L of a structure can be a design choice for control-ling thermal conductivity. The effect of L has also been a richtopic for fundamental insight, revealing information about themean free paths k of dominant phonons and the transition betweenparticle- and wavelike transport. Equivalently, results have oftenbeen reported in terms of interfacial density. We compile selectedexperimental results in Fig. 3, which shows that trends with Lhave been much more difficult to interpret than trends with T. Inthe limit of thick layers, bulk resistances dominate interfacialresistances, and k should be independent of L (i.e., independent ofthe interface density). As L decreases, a transition toward the bal-listic regime corresponds with increased significance of interfacialresistance, and k should decrease with shrinking L. Several earlyexperiments reported this trend down to the shortest periodlengths, corresponding to the persistence of particlelike phonontransport [109,115,120,121]. Among these, the trend observed byCapinski et al. [120] does not reflect the coherence suggested bythe temperature dependence in the same report, which highlightsthe complexity of the research problem. More recently, Koh et al.[123] showed a persistence of particlelike transport in AlN/GaNsuperlattices, dominated by strongly wavelength-dependent inter-face scattering. This behavior corroborates theoretical predictionsof L dependence using the BTE [83–85].

Other experimental reports, however, have suggested a transi-tion to wavelike transport at very short L, as signified by anonmonotonic trend that gives a minimum in k at short L. Simkinand Mahan [89] predicted this trend theoretically for coherent sys-tems, showing that k should exhibit a minimum around L � k,marking a transition between particlelike transport for thickerL and wavelike transport for shorter L. This was the proposed

Fig. 2 Selected experimental reports of thermal conductivityin multilayers and superlattices as a function of temperature T .We list the period length of each system in nanometers. Theorypredicts that incoherent transport should exhibit a plateau withrising T , while coherent transport should exhibit some inversetendency due to mini-umklapp scattering at high T .

Fig. 3 Experimental measurements of thermal conductivity inmultilayers and superlattices as a function of period length L.All data are selected around 300 K. As a tuning parameter, Lseems to allow control of k over a factor of up to 3 or 4. Data forcross comparison with Fig. 2 are available for Lee et al. [109],Capinski et al. [120], and Costescu et al. [122]. Theory predictsthat a monotonic increase with L indicates incoherent trans-port, but a “minimum conductivity” at short periods indicates atransition to coherent transport.



explanation for early measurements of conductivity by Venkata-subramanian in Bi2Te3/Sb2Te3 superlattices [110] and by Chakra-borty et al. in Si/Ge [116], as motivated by thermoelectrics. Insupport of those results, Wang et al. [124] measured sound speedsin Bi2Te3/Sb2Te3 superlattices that were 10% lower than pre-dicted from impedance alone, suggesting a lowered group velocityassociated with coherence. L dependence has also indicated coher-ent transport in some more recent conductivity measurements, asreported by Zhang et al. for InGaAs/InGaAsP [117] and by Tonget al. for Ge2Te3/Sb2Te3 [119].

These compiled results show that the identification of dominantmechanisms in superlattice and multilayer transport is a complexproblem. At this point, it appears plausible that phonon coherencecan have a significant effect on thermal transport, but only in cer-tain systems. Superlattices of III-V materials, for example, appearto be better candidates for significant coherent transport thangroup IV superlattices. However, the apparent importance ofcoherent effects has varied even among reports for the same mate-rial system; for example, Refs. [109,112,116] for Si/Ge super-lattices and Refs. [110,121] for Bi2Te3/Sb2Te3. This implies theintuitive result that the importance of different suppression mech-anisms is very sensitive to the particular growth quality of eachstructure.

As a design parameter for controlling thermal conductivity, theperiod length is important when interfacial resistances are impor-tant (compared with bulk resistances from alloy or bulk defectscattering; for example). In the reports with greatest sensitivity toL, the attainable conductivities range across a factor of 3 or 4 nearroom temperature. This is the case for the reports by Capinskiet al. (about 4 to 14 W m�1K�1) [120], by Venkatasubramanian(0.2 to 0.6 W m�1K�1) [110], by Costescu et al. (0.5 to1.5 W m�1K�1) [122], and by Zhang et al. (2 to 6.6 W m�1K�1)[117].

3.5 Nonideal Interfaces in Multilayers. Nonideal interfaceshave been the primary explanation for the difficulty in observingcoherent transport in superlattices. In a set of molecular dynamicssimulations, Daly et al. [95] demonstrated a clear minimumthermal conductivity as a function of period length, as long as theinterfaces were perfect. Interfaces with random, single-atomroughness—confined to only one monolayer at each interface—caused the minimum to disappear, indicating that the slight disor-der had disrupted coherent phonon transport. However, a simplis-tic model was used for GaAs/AlAs, in which a single “average”atom represented each two-atom primitive cell. More recently,lattice dynamics calculations by Ren et al. [90] and MD simula-tions by Landry and McGaughey [99] demonstrated the sameeffect in diamond-structure Si/Ge superlattices, in which a simi-larly small extent of interfacial mixing destroyed coherent effects.

These predictions suggest that coherent effects may not be animportant suppression mechanism in experimental systems witheven single-monolayer roughness or interdiffusion. However, onesignificant shortcoming of simulations to date is that their in-planedimensions have been limited by computational resources to tensof angstroms (i.e., the periodicity enforced in those dimensions).This constrains the correlation length of simulated roughness, asdescribed in Sec. 2.5 for single interfaces. In contrast, monolayersteps at a GaAs/AlGaAs interface grown by molecular beam epi-taxy (MBE) may be separated in-plane by hundreds of angstroms,with broad, atomically smooth “islands” in between [125]. Experi-mentally, Duquesne [118] used growth temperature to explicitlyvary the cross-plane extent of interfacial roughness d0 in GaAs/AlAs superlattices and measured the effect on thermal conductiv-ity by 3x. Indeed, the data showed no discernible differencebetween the sample with “perfect” interfaces (d0 ¼ 0 nm) and thesample with one-monolayer steps about the interfaces (d0 ¼ 0:15nm), in apparent contrast with predictions. Broader interfaces,d0 ¼ 0:45 nm, did reduce conductivity. Thus, the disruptiveeffects of roughness and mixing in simulations, currently

constrained in the in-plane dimension, may not apply directly toreal superlattices with broad islands. An interesting recent studyby Termentzidis et al. [126] sought to capture these effects in sim-ulations, revisiting the simplified fcc model for GaAs/AlAs usedby Ren et al. while varying both the height and width of interfacialislands. The results reproduce the expected wavelike behaviorwith perfect interfaces and particle behavior with certain rough-ness, but some unexpected behavior with other rough interfaces.Hence, further work is still required to dissect the problem.

Lattice mismatch has also been shown to play an important rolein superlattice thermal conductivity. For example, the Si/Ge sys-tem exhibits a mismatch around 4%, and dislocations will formwhen layers are sufficiently thick. Lee et al. [109] invoked thisphenomenon to explain an anomalous dropoff in experimentalmeasurements of k as the period length L increased past about100 A. Even without the formation of dislocations, lattice mis-match causes interfacial lattice strain that could potentially affectoverall transport. MD simulations by Chen et al. [96] demon-strated a minimum in k versus L for perfect interfaces, but thetrend disappeared when the two materials had a 4% lattice mis-match, despite the constituent layers being too thin to form dislo-cations. The authors suggested that the mere presence of strainmust cause sufficiently diffusive interfacial scattering to disruptcoherence.

3.6 Multilayers: Applications and Outlook. Although pho-non transport in multilayers and superlattices still holds greatpotential for further basic research, researchers have gleanedenough insight to apply to the active design of some novel struc-tures. Taken together, these insights show that coherent effectsmight be important in certain systems, but in practical terms, thedirect reduction by interface scattering is easier to exploit. Wenote two dramatic implementations in reports of fully dense solidsdesigned for ultralow thermal conductivity. Costescu et al. [122]used atomic layer deposition and magnetron sputtering to growmultilayers of W and amorphous Al2O3, which showed theexpected decrease of k with interface density and a minimum karound 0.6 W m�1K�1. With a similar approach, Chiritescu et al.[127] measured a conductivity of around 0.05 W m�1K�1 acrosslayers of WSe2, which were themselves crystalline but stackedwith random in-plane orientation. The resulting cross-plane con-ductivity was at least an order of magnitude lower than in typicalamorphous solids [128] due to weak interlayer van der Waalsbonding: conductivity actually increased when the layered struc-ture was damaged by ion irradiation. The WSe2 system is perhapsnot a multilayer in the typical sense, but it certainly illustrates thestrategy of using laminar structures to dramatically suppress ther-mal transport.

As work continues on multilayer systems, we note that futureresearch should carefully evaluate contributions from size effects,which may be potentially conflated with contributions from thelayered structure. Cheaito et al. [129] reported measurements ofthermal conductivity in nondilute Si1�xGex alloys, showing thatsize effects (i.e., boundary scattering) actually dominated overalloy scattering, even in films hundreds of nanometers thick.Hence, when evaluating the mechanisms affecting phonon trans-port in multilayer structures, and particularly when comparingtheir thermal conductivity to the “alloy limit,” the total samplethickness is likely to have a more important effect than previouslyrecognized.

Thermoelectric materials extend the problem of minimizingthermal conductivity by also requiring high electrical conductiv-ity, corresponding to an ideal material that behaves simultane-ously as an “electron crystal and phonon glass.” Multilayeredstructures are one approach for attaining such a material, if theinterfaces can be designed to preferentially scatter phonons butnot electrons. An interesting illustration of this concept is the“isotopically modulated” 28Si/29Si superlattice, in which Brachtet al. [130] measured a cross-plane thermal conductivity around



one-third that of bulk silicon (with L � 10 nm). In addition to in-terest in its inherent properties, the structure also represents aunique analog to common MD simulations of purely mass-variedmaterials. However, no direct measurement of thermoelectricperformance was made. Examples of periodic structures withhigh reported thermoelectric figures of merit ZT include Bi2Te3/Sb2Te3 thin film superlattices [131] and PbSeTe/PbTe quantumdot superlattices [132]. The latter example begins to blur our defi-nition of a superlattice, and suggests the broad range of structuresthat are being studied for thermoelectric applications. In a laterexample, Kim et al. [133] embedded ErAs nanoparticles of vari-ous size and spacing in In0:53Ga0:47As alloy, demonstrating a sys-tematic reduction in k with corresponding increase in predictedZT. Nanocomposite materials are also very promising for thisapplication, with suppressed phonon transport due to high interfa-cial density as in superlattices. For a detailed discussion of nano-composites for thermoelectrics, we refer to the review by Minnichet al. [134].

Phase change memory (PCM) is yet another application forengineering reduced thermal conductivity. PCM uses varyingpulses of electrical current to switch the state of the recording me-dium between crystalline and amorphous phases. One strategy toreduce the necessary programming current has been to engineerphase change materials with lower thermal conductivity. Forexample, compared to typical GeSbTe-type (GST) alloy, Chonget al. [135] demonstrated reduced programming currents in cellsbased on GeTe/Sb2Te3 superlattices, which they attributed largelyto reduced thermal conductivity. Tong et al. [119] used 3x meas-urements and MD simulations to verify the reduction in thermalconductivity with increasing interfacial density. However, wenote that many factors contribute to PCM performance, andthermal properties are only one aspect. Simpson et al. [136] alsostudied GeTe/Sb2Te3 superlattices and demonstrated significantlyimproved performance over GST, but thermoreflectance measure-ments showed that thermal conductivity suppression was not amajor contributor. Rather, they attributed the performance gainsto a minimized difference in configurational entropy between thecrystalline and amorphous states, as effected by the layered struc-ture. As the understanding of superlattice properties develops inseparate fields, this example demonstrates the importance of moreinterdisciplinary work. This example also highlights the futurepotential for synergistic design, in which a system could poten-tially exploit thermal, electrical, and other effects simultaneouslyin a single structure, especially at multiple scales.

In the future, even further reductions to thermal conductivitymight be attained by precisely manipulating the periodicity (orlack thereof) in multilayer structures. Simulations by Landry et al.[98] and by Frachioni and White, Jr. [102] both suggest that struc-tures with irregular layer thicknesses, while maintaining sharpinterfaces, could exhibit conductivities significantly lower than inregular periodic structures. To our knowledge, this strategy hasnot been explored in systematic experiments. The same strategywas proposed by Lau et al. [137] for suppressing photonic thermaltransport, predicting that coherence in an aperiodic system maysuppress conductance even more strongly than interface-dominated incoherent transport. Given the findings to date thattransport in superlattices and multilayers depends so strongly oninterfacial quality, detailed characterization accompanying experi-mental results is crucial to further understanding. Photolumines-cence and Raman scattering are examples of tools that have beenused for characterization of interface roughness in superlattices,but not often in direct conjunction with thermal measurements[118]. There are also examples of using transmission electronmicroscopy to glean useful information about interfaces in super-lattice structures [138,139]. Experimental characterization ofinterfaces could be especially useful when coupled with modeling.Advances in implementations of first-principles methods such asdensity functional theory (DFT) are enabling modeling of morerealistic interfaces, which may help to resolve their effect oncoherent transport in experimental measurements.

The question of electron-phonon coupling across metal/insula-tor interfaces, raised briefly in Sec. 2.8 for single interfaces, isalso receiving increasing attention in multilayer systems. Forexample, very recent reports by Li et al. [140] and Bozorg-Grayeliet al. [141] have investigated the thermal transport in metal/semi-conductor Mo/Si multilayers, which have optical properties desir-able for extreme ultraviolet lithography. Detailed modeling ofsuch systems is sparse; one example is a report by Mahan [142] inwhich a continuum-level description was used to calculate cross-plane conductivity. Again, this topic is beyond the scope of thisreview, but we expect that it will be an area of increasing futureresearch.

Future work could also investigate superlattices in reduced-dimensionality structures. A decade ago, Li et al. [143] reportedtemperature trends in the conductivity of Si/SiGe nanowires withdiameters as small as 22 nm, finding alloy scattering to be domi-nant. Much potential for experimental work remains, and modelshave indicated interesting effects in low-dimensional superlatti-ces. For example, Shiomi and Maruyama [144] simulated the con-ductivity of isotopically modulated 12C/13C nanotubes, showing aslight minimum in k versus L. However, the predicted reduction isrelatively small (�40%–50% compared to pure 12C nanotube)given the anticipated difficulty in fabricating such a system. Morerecently, Jiang et al. [145] predicted the conductivity of nanorib-bon BN/graphene superlattices, which may be somewhat morefeasible to produce experimentally. Using a nonequilibriumGreen’s function approach, they also reported a minimum withperiod length. However, in both the nanotube (1D) and nanorib-bon (2D) systems, the minimum k occurred with period lengthsL� k, apparently ruling out the transition between particle- andwavelike transport as a cause. The trend is not yet well under-stood, but suggests that superlattices with reduced dimensionalitycould be a source of even more interesting transport phenomenarelated to the control of mode confinement.

4 Polyjunctions

We have discussed single interfaces and multilayer materials aswell as how their thermal properties scale with various effectssuch as material composition, temperature, and structure. Nano-structures that lie between the limits of a single interface and theperiodic interfaces of a multilayer present additional challenges interms of modeling. Not only will the interfacial conditions such asroughness and defects influence transport, but the wave nature ofphonons must also be considered as in multilayers. Great stridesin the understanding of interfacial and multilayer transporthave been made in the past decade, and we anticipate thatlessons learned thus far will inform modeling approaches tothis new regime. We refer to this class of nanostructures as“polyjunctions” owing to the number of materials or interfacesthat participate and the engineering purpose of such structures: tojoin materials while giving increased control over transportproperties. The generic structure of a polyjunction is presented inFig. 4; it consists of two bulk materials joined by N materiallayers that are on the order of angstroms to nanometers in thick-ness and selected in order to tune the overall phonon conductance.The thinness of the layers compared to phonon wavelength andmean free path ensures that the system behaves like one indivisi-ble structure instead of a series of isolated interfaces. Here we

Fig. 4 Schematic diagram of a polyjunction. The material andthicknesses of the N layers are selected to tune the transport.For N ¼ 0, the original interface is recovered.



review two configurations: one where a polyjunction joins mis-matched materials, and one where it is introduced into a singlebulk material.

4.1 Polyjunction Bridge. It is evident from Eqs. (3) and (4)and the results discussed in Sec. 2.1 that phonon transmissivity ishigher between matched materials. Thus, a polyjunction can beused to split an interface between mismatched materials into two(or more) interfaces between better matched materials in anattempt to raise the overall hBD. Liang and Tsai [146] investigatedthe effects on hBD of varying the inserted film bond strength andthickness in a simulation. They found that, for weakly bondedfilms, a monolayer of the film material provides enhancement,while any more than this will lead to a decrease in hBD. Con-versely, for a thin film with strong bonds, enhancement is seenacross all thicknesses with the maximum values dependent on thebond strength. The degree of enhancement was also dependent onthe mismatch between the bulk materials, with a smaller mismatchleading to less enhancement after the thin film was added. In sub-sequent simulations, Liang and Tsai [147] found that the conduct-ance was maximized for a film having the geometric mean of theDebye temperatures of the interface solids. Overall, a maximumenhancement factor of about 2.5 in hBD was observed. Simulationsby English et al. [35] also showed that hBD could be enhancedwith the addition of a thin film and that the film should be as thinas possible and have vibrational properties intermediate to thebulk materials. The authors also demonstrated that adding moder-ate interdiffusion further enhanced hBD. Using these methods, hBD

was ultimately enhanced by as much as 53% over the originalinterface. The success of polyjunctions in the enhancement of hBD

is predicated on the splitting of the interfaces compensating forthe increased resistance from additional material. From a model-ing perspective, the thinness of the films makes it unclear howphonons of different wavelengths and mean free paths will react.It is likely that the phonon population is in a highly nonequili-brium state within the polyjunction.

Polyjunction architectures have been investigated for more thanone inserted film. The addition of more films at the interfaceoffers greater opportunity for tuning the thermal transport. In asimulation study, Le et al. [148] demonstrated that for a single,long wavelength mode with normal incidence, the effective trans-missivity could be enhanced by 7.3 percentage points for a N ¼ 2polyjunction or 8.8 percentage points using a N ¼ 4 polyjunctionover the baseline interface by varying the layer masses. Further-more, the authors found that the transmissivity was relativelyinsensitive to the precise vibrational properties of the films, aslong as they bridged the vibrational gap in close to a geometric se-ries of layer mass. This is encouraging for real world applicationswhere the selection of materials is dependent on many practicalconsiderations, limiting the control over the exact vibrationalproperties within the polyjunction. The polyjunction films mayalso be given alternating masses, reminiscent of a superlattice, inorder to minimize the transmission from one material to another[148,149]. Using this method, Le et al. [148] showed a 23.4 per-centage point reduction to the transmission for a N ¼ 4 polyjunc-tion over the baseline interface. Increasing the number of layersdecreases the transmissivity, but this effect quickly saturates afterabout ten layers for long wavelengths. The saturation can beattributed to coherent effects, i.e., for regular period length andmass variation, interference among transmitted and reflected pho-nons at different points in the structure effectively prevents furtherreflection, indicating superlatticelike transport [149]. The satura-tion can be delayed by disrupting the periodic nature of the poly-junction; thus eliminating some of the coherent interference thatled to the saturation. The disruption can be achieved by varyingthe layer thickness [149] or by varying the layer masses [148], forinstance. Enacting these measures led to a further reduction intransmissivity of 3.1 percentage points for the N ¼ 4 polyjunction[148]. Li and Yang [150] examined the spectral transmission

dependence across superlatticelike polyjunctions using a Green’sfunction approach. Their results corroborate the mechanism ofsaturation and that after about N ¼ 10, the polyjunction acts like abulk superlattice.

4.2 Polyjunction Barrier. Inserted thin films may also beapplied to an originally single material [38,151–153]. In thiscapacity, the polyjunction acts as a sort of phonon “speed bump.”In wavepacket studies of a LJ/heavy-LJ/LJ system, Tian et al.[152] found that in the long wavelength limit, the transmissionwas described by the AMM with consideration for constructive/destructive interference effects. For higher wavenumbers, wherethe barrier material did not contain energy levels that would allowelastic scattering, the wavepacket could tunnel through the barriervia evanescent wave transmission. That is, treating the thin film asa bulk material, the thin film contained complex wavevectors ofthe correct frequency, leading to exponentially decreasing packetamplitude that survived to the second interface due to the thinnature of the film [6,152]. Landry and McGaughey [151] studied asimilar system composed of Si/Ge/Si. They found that the con-ductance decreased with film thickness until a thickness of about2 nm. From lattice dynamics calculations, they found that for thinGe films, the density of states differed considerably from bulk Ge,with states at frequencies above the cut off frequency of bulk Ge.As the film thickness was increased, the density of states returnedto the bulk values, indicating that the thickness plays a rolebeyond determining conditions for interference [151]. Huang andChang [153] expanded on this geometry by modeling the effectsof Ge quantum dots grown on the Ge thin film. The thermalboundary conductance was found to be a nonmonotonic functionof the quantum dot density, achieving a maximal value at a den-sity of about 0.25 surface coverage fraction. Just as with theN ¼ 1 polyjunction studied by English et al. [35], this result dem-onstrates that the interface condition of roughness can be used tofurther tune polyjunction transport. One application of the poly-junction phonon “speed bump” is in phonon engineering; e.g., thethin film acts as a low pass filter [152] or as a method for inducingmode conversion. The primary method of heat generation in tran-sistors is the scattering of electrons with the lattice to produceoptical phonons [1]. The slow speed of optical phonons creates athermal bottleneck. If polyjunctions could be engineered to pro-mote an optical to acoustic mode conversion, then the heatingnear the transistor could potentially be reduced.

4.3 Polyjunction Outlook. Polyjunctions offer a unique andpromising approach to tuning thermal boundary conductance.Although it has not been experimentally demonstrated, it is likelythat polyjunctions will also exhibit enhancements in thermalboundary conductance of real systems. Polyjunctions with a singlelayer are similar in concept to SAM junctions (Sec. 2.3), with avital difference being the much larger number of accessible vibra-tional modes. This should make possible larger magnitudes inthermal boundary conductance, if not greater potential for tuningas well. This approach to tuning thermal boundary conductance isstill in its infancy; any reasonable future investigations will aidour understanding. In particular, rigorous experimental observa-tion of enhanced (and decreased) thermal boundary conductanceusing a polyjunction approach will validate the technology andencourage further research.

5 Conclusions

In conclusion, we have reviewed the mechanisms of phonontransport across interfaces, multilayers, and polyjunctions in lightof recent theoretical, simulated, and experimental findings. Foreach type of feature, we have emphasized opportunities for tuningthermal transport, which may be harnessed to improve the thermalperformance of structures with high interface densities. Realisti-cally, the solution to current thermal challenges lies not with one



specific approach, but will require device design that simultane-ously incorporates thermal transport with other performance crite-ria. In addition, successful strategies for thermal dissipation orinsulation in modern technologies will require the engineering ofthermal transport at all available length scales from the bottom up,starting with the nanoscale issues described here.

Acknowledgment

The authors would like to acknowledge financial support fromthe Air Force Office of Scientific Research (Grant No. FA9550-09-1-0245). C. H. B. is grateful for support from the Departmentof Defense through a NDSEG Fellowship and the Virginia SpaceGrant Consortium through a Graduate STEM Research Fellow-ship. N. Q. L. is grateful for support from the UVa Office of theVice President for Research. The authors would like to thank Pat-rick E. Hopkins for insightful discussion.

Nomenclature

D ¼ spectral phonon density of states, m�3 (rad/s)�1

L ¼ multilayer period length, mT ¼ temperature, KZ ¼ acoustic impedance, kg m�2s�1

a ¼ lattice constant, md ¼ layer thickness, m

d0 ¼ extent of interfacial roughness, mf ¼ Bose–Einstein distribution function�h ¼ reduced Planck constant, J s

hBD ¼ thermal boundary conductance, W m�2K�1

k ¼ thermal conductivity, W m�1K�1

q ¼ heat flux, W m�2

r ¼ interatomic distance, mu ¼ interatomic interaction energy, Jv ¼ phonon group velocity, m s�1

Greek Symbols

a ¼ phonon transmission probabilitye ¼ Lennard-Jones energy scale, Jj ¼ effective spring constant, N/mk ¼ phonon mean free path, ml ¼ cosine of phonon incidence angler ¼ Lennard-Jones length scale, mht ¼ zenith angle of transmitted phonon, radx ¼ angular frequency, rad s�1

xc ¼ cut off frequency, rad s�1

Subscripts

j ¼ phonon branch index1,2 ¼ material index

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Tuning Phonon Transport: From Interfaces to Nanostructures