applied physics reviews nanoscale thermal … physics reviews nanoscale thermal transport david g....

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 2 15 JANUARY 2003

APPLIED PHYSICS REVIEWS

Nanoscale thermal transportDavid G. Cahilla)

Department of Material Science and Engineering and the Frederick Seitz Materials Research Laboratory,University of Illinois, Urbana, Illinois 61801

Wayne K. FordIntel Corporation, 5200 NE Elam Young Parkway, Hillsboro, Orgeon 97124

Kenneth E. GoodsonDepartment of Mechanical Engineering, Stanford University, Palo Alto, California 94305

Gerald D. MahanDepartment of Physics, Pennsylvania State University, University Park, Pennsylvania 16802

Arun MajumdarDepartment of Mechanical Engineering, University of California, Berkeley, California 94720

Humphrey J. MarisDepartment of Physics, Brown University, Providence, Rhode Island 02912

Roberto MerlinDepartment of Physics, University of Michigan, Ann Arbor, Michigan 48109

Simon R. PhillpotMaterials Science Division, Argonne National Laboratory, Argonne, Illinois 60439

~Received 28 January 2002; accepted 1 August 2002!

Rapid progress in the synthesis and processing of materials with structure on nanometer lengthscales has created a demand for greater scientific understanding of thermal transport in nanoscaledevices, individual nanostructures, and nanostructured materials. This review emphasizesdevelopments in experiment, theory, and computation that have occurred in the past ten years andsummarizes the present status of the field. Interfaces between materials become increasinglyimportant on small length scales. The thermal conductance of many solid–solid interfaces have beenstudied experimentally but the range of observed interface properties is much smaller than predictedby simple theory. Classical molecular dynamics simulations are emerging as a powerful tool forcalculations of thermal conductance and phonon scattering, and may provide for a lively interplayof experiment and theory in the near term. Fundamental issues remain concerning the correctdefinitions of temperature in nonequilibrium nanoscale systems. Modern Si microelectronics arenow firmly in the nanoscale regime—experiments have demonstrated that the close proximity ofinterfaces and the extremely small volume of heat dissipation strongly modifies thermal transport,thereby aggravating problems of thermal management. Microelectronic devices are too large toyield to atomic-level simulation in the foreseeable future and, therefore, calculations of thermaltransport must rely on solutions of the Boltzmann transport equation; microscopic phonon scatteringrates needed for predictive models are, even for Si, poorly known. Low-dimensional nanostructures,such as carbon nanotubes, are predicted to have novel transport properties; the first quantitativeexperiments of the thermal conductivity of nanotubes have recently been achieved usingmicrofabricated measurement systems. Nanoscale porosity decreases the permittivity of amorphousdielectrics but porosity also strongly decreases the thermal conductivity. The promise of improvedthermoelectric materials and problems of thermal management of optoelectronic devices havestimulated extensive studies of semiconductor superlattices; agreement between experiment andtheory is generally poor. Advances in measurement methods, e.g., the 3v method, time-domainthermoreflectance, sources of coherent phonons, microfabricated test structures, and the scanningthermal microscope, are enabling new capabilities for nanoscale thermal metrology. ©2003American Institute of Physics.@DOI: 10.1063/1.1524305#

a!Electronic mail: [email protected]

7930021-8979/2003/93(2)/793/26/$20.00 © 2003 American Institute of Physics

Downloaded 18 Nov 2008 to 171.64.49.29. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

79

6

6

mllthicoomecymhthietaceecd

cumhtste

alre-all

me-ns.ven

ve100the

ons,This

be-na-

ofre-ep-onfp.of

iffi-a-re-t areron, thehatthe

ofntal

ns-t aport

el-s of

r-aticstill

alri-dif-thet inurest in, isme-

794 J. Appl. Phys., Vol. 93, No. 2, 15 January 2003 Appl. Phys. Rev.: Cahill et al.

TABLE OF CONTENTS

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 794II. HEAT TRANSPORT AND PHONON

DYNAMICS AT INDIVIDUAL INTERFACES. . . 795A. Thermal conductance of model interfaces. .. . . . 795B. Heat pulse and coherent phonon experiments...C. Molecular-dynamics simulation of heat

transport and phonon dynamics. . . . . . . . . . . . . . 797D. What is temperature?. .. . . . . . . . . . . . . . . . . . . . 799E. Transport theory. . . . . . . . . . . . . . . . . . . . . . . . . . 800

III. PHONONS AND THERMAL TRANSPORT INNANOSCALE STRUCTURES AND DEVICES.. 801

A. Silicon films and devices. . . . . . . . . . . . . . . . . . . 8011. Conduction along semiconducting

monolayers. . . . . . . . . . . . . . . . . . . . . . . . . . . 8012. Monte Carlo solutions of the Boltzmann

transport equation. . . . . . . . . . . . . . . . . . . . . . 8023. Nanoscale hotspots in devices. . . . . . . . . . . . 803

B. Nanostructures with tailored phonon transportproperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804

C. Phonon transport in 1D nanostructures. . . . . . . . 804D. Nanoscale 3D conduction and mass transport

in polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805IV. THERMAL TRANSPORT IN

NANOSTRUCTURED MATERIALS. . . . . . . . . . 806A. Amorphous and nanoporous materials. . . . . . . . 806B. Multilayers and superlattices. . . . . . . . . . . . . . . . 808C. Superlattices: Theory and computation. . . . . . . . 811

V. NEEDS AND LIMITS OF METROLOGY. .. . . . . 812A. Time-domain Thermoreflectance. . . . . . . . . . . . . 812B. Microfabricated probes. . . . . . . . . . . . . . . . . . . . . 813C. Coherent optical methods. . . . . . . . . . . . . . . . . . . 815

VI. OUTLOOK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

I. INTRODUCTION

Recent advances in synthesis, processing, andcroanalysis are enabling the routine production of wecharacterized materials with structure that varies onlength scale of several nanometers. Examples are semductor quantum dots and superlattices, polymer nanocposites, multilayer coatings, microelectronic and optoeltronic devices, and microelectromechanical sensors. Manthese nanoscale structures already have important comcial applications, while others are studied scientifically. Tmost important devices, in terms of worldwide sales, areintegrated circuits used in computer circuits and memorDevice engineers have been astonishingly clever at maining Moore’s Law, which states that the individual devicomponents get smaller every year. Soon the field-efftransistors~FET! will have a channel length of 100 nm, an50 nm devices are projected in the near future. In mostrent and envisioned applications of nanostructures, thermanagement is a serious issue. In some devices, succomputer processors or semiconductor lasers, one wanget the heat away as efficiently as possible—these sys

Downloaded 18 Nov 2008 to 171.64.49.29. Redistribution subject to AIP

6

i--en--

-ofer-ees.in-

t-

r-alasto

ms

need high thermal conductivity. In others, such as thermbarriers or thermoelectric materials used for solid-statefrigeration, one wants the thermal conductivity to be as smas possible.

Our discussion emphasizes thermal transport in nontallic systems, in which heat is transported by phonoPhonons have a wide variation in frequency, and an elarger variation in their mean-free-paths~mfps!. However,the bulk of the heat is often carried by phonons of large wavector, and they have mfps at room temperature of 1–nm. So in many systems of current interest, the scale ofmicrostructure is the same scale as the mfp of the phonand sometimes comparable to the phonon wavelength.clearly necessitates an understanding of heat transportyond that achievable at the continuum level. So far, no alytical theories have adequately treated the wave naturephonons that will be manifest at these length scales. Moover, this equality in length scales raises important conctual questions, among which is how to define temperaturethe length scale of the phonon wavelength and phonon m

This review will summarize the present understandingthermal transport in nanostructures. The experimental dculties begin with the severely limited capabilities for mesuring thermal transport in increasingly small systems. Acent review summarizes the experimental techniques thabeing used to probe thermal conduction at a submiclevel.1 These methods are also discussed here. Currentlyonly method with nanometer-scale spatial resolution tmay be applied to nanometer structures is based onatomic force microscope~AFM!. However, AFM measure-ments are only beginning to provide an understandingthermal transport in nanoscale structures; the experimefield is certainly in its infancy.

The theory and simulation of nanoscale thermal traport is at a similarly immature stage. The recognition thamore detailed theoretical understanding of heat trans

than achievable using Fourier’s Law of heat conduction,Q

52k¹T whereQ is the heat flux,k the thermal conductiv-ity and ¹T the temperature gradient, has lead to the devopment of approaches based on the numerical solutionthe Boltzmann transport equation~BTE! and to atomic-levelsimulations of thermal conductivity. Each still requires futher methodological developments and the first systemapplications of these methods to heat transport issues arebeing refined.

Since interfaces play a critical role in nanoscale thermtransport, we begin our discussion with a review of expements and theory on the transport of heat between twoferent materials or between twin or twist boundaries ofsame material. In Sec. III, we focus on thermal transpormicrofabricated nanostructures and synthetic nanostructof reduced dimension. The related topic of heat transpornanostructured thin film materials, including superlatticesaddressed in Sec. IV. We explore issues associated withtrology and measurement science in Sec. V.

license or copyright; see http://jap.aip.org/jap/copyright.jsp

rstonims

ctlwctht-pnt

ura

Be

ongpth

en

clt

heon

ofalruaferonthcr

tylascin

al

in-ter-free-t insi-in--mal

eenin

ghtsu-

izaof-heeenanderin

thandi-stan-

useennd

xis-thealsnt

n anerem-was

t bede-s-an

canthisrts,n ofderre-

b-on

thetwoof

795J. Appl. Phys., Vol. 93, No. 2, 15 January 2003 Appl. Phys. Rev.: Cahill et al.

II. HEAT TRANSPORT AND PHONON DYNAMICS ATINDIVIDUAL INTERFACES

The idea that an interface should produce a thermalsistance is intuitively appealing, since an interface contutes an interruption in the regular crystalline latticewhich phonons propagate. For an interface between disslar materials, the different densities and sound speeds rein a mismatch in the acoustic impedances; this is direanalogous to the mismatch in the refractive indices of toptically different materials. The effect that this impedanmismatch has on phonon transmission is captured inacoustic-mismatch~AM ! model. By assuming that no scatering takes place at the interface, and by imposing approate stress and displacement boundary conditions at the iface, the AM model gives the transmission coefficienttAB forphonon energy in materialA incident normal to the interfacewith materialB as:

tAB54ZAZB

~ZA1ZB!2 , ~1!

whereZ5rc is the acoustic impedance, withc andr beingthe speed of sound and mass density, respectively.2,3 In theAM model, the interface has no intrinsic properties bmerely joins the two grains. As a consequence a twist gboundary, in which the normals on the two sides of the Gare crystallographically identical, would incorrectly be prdicted to have a transmission coefficient of one.

By contrast, in the diffuse mismatch~DM! model it isassumed that all phonons striking an interface loses memof where they came from. Then, the probability of beiscattered to one side of the interface or the other is simproportional to the phonon density of states. Thus, as inAM model, the fraction of energy transmitted is independof the structure of the interface itself.3

While the above two models provide useful referencalculations against which to compare experimental resuand for many cases give rather similar predictions, neitcaptures the complexity of the interaction between phonand real interfaces.

As we will see in this section, the interactionsphonons with a single interface still offers significant chlenges to both experiments and theory/simulation. We fireview experiments on thermal transport across individinterfaces, including both the classic Kapitza problem osuperfluid-helium/solid interface and model solid/solid intfaces. We point out that spectral methods, involving phonof well-defined frequencies and wave vectors, offerpromise of providing insights into thermal transport not acessible to more traditional experiments. We summarizecent atomic-level simulations of thermal conductivithrough individual interfaces and identify a need for simutions of the interaction of individual phonons with interfaceFinally, we point out that the simulations of thermal condutivity raise fundamental thermodynamics issues concernthe definition of temperature of nonequilibrium nanoscsystems.


e-i-

i-ultyoee

ri-er-

tin

-

ry

lyet

es,rs

-stl

a-s

e-e-

-.-g

e

A. Thermal conductance of model interfaces

We consider the thermal conductance of an isolatedterface, i.e., an interface that is separated from other infaces by a distance that is large compared to the mean-path of the lattice vibrations that dominate heat transporthe material. In this limit, we can ignore coherent superpotion of lattice waves reflected or transmitted by adjacentterfaces. Swartz and Pohl3 give an exhaustive review of thermal boundary resistance, the inverse of interface therconductance, through 1988.

The existence of a thermal boundary resistance betwa solid and superfluid helium was first detected by KapitzaRef. 4. When the discovery was first made, it was thouthat this resistance was due to some special property ofperfluid helium. Only later was it realized that such a Kaptresistance,RK , exists at the interface between any pairdissimilar materials. Khalatnikov2 developed the acousticmismatch model to explain this thermal resistance. Tacoustic mismatch is extremely large at an interface betwhelium and most ordinary solids because the densitysound velocity of helium are very much less than for othmaterials. While Khalatnikov’s theory was able to explathe existence of the Kapitza resistance, the value ofRK thatwas calculated was between 10 and 100 times largertypical experimental values measured at 1–2 K. This incates that phonons are able to cross the interface subtially more easily than expected.

Remarkably, the reason for this so-called ‘‘anomalotransmission’’ at these low temperatures has still not bestablished. It was proposed by Challis, Dransfeld, aWilks5 that the transmission was increased due to the etence of a compressed layer of helium on the surface ofsolid. This layer is formed because of the van der Waattraction of helium atoms to a solid surface. An importaexperiment was performed by Weberet al.6 They measuredthe transmission of phonons across the interface betweealkali–halide crystal and helium. Very clean surfaces wproduced by cleaving the alkali-halide crystals at low teperature. For these surfaces no anomalous transmissionobserved, indicating that the excess transmission musassociated with some sort of surface contamination orfects. Olson and Pohl7 verified that the anomalous transmision between Si and liquid helium directly correlates withincrease in the diffuse phonon scattering.

Since the work of Weberet al.,6 a number of attemptshave been made to explain how surface contaminationhave such a large effect on the transmission. Much ofwork is reviewed in Refs. 3, 8, and 9. Despite these effono generally accepted theory has emerged. The resolutiothis problem requires more detailed experimental work uncontrolled conditions with well-characterized surfaces ppared and studied under ultrahigh vacuum.

Of course, while not directly relevant to technical prolems of thermal management in nanostructures, the worksuperfluid helium does leave open the possibility thattransmission of phonons across the interface betweensolids may be substantially affected by a small number


y

odafn

nd

eT

ini

teurt

exli

thebal-dyiza-theith

ular.ing.

and. A

e of

re-rial.to

:

tal

fce

le in

to 2la-axi-


surface defects. The extent to which this occurs has notbeen established.

Some recent data for the thermal conductances of msolid/solid interfaces are shown in Fig. 1. Surprisingly, neroom temperature, the highest thermal conductancemetal/dielectric interfaces measured by picosecothermoreflectance,10 Ti/Al 2O3 and Al/Al2O3 , is only a factorof ;5 larger than the lowest conductance, Pb/diamoOther data for metal/dielectric interfaces,3,11–13for the mostpart, fall between these two extremes.

B. Heat pulse and coherent phonon experiments

Phonon transmission across a single interface has bstudied using both coherent and incoherent phonons.study incoherent phonons a pulse of energy is depositedthe material on one side of the interface. This generatescoherent phonons with a frequency spectrum that is demined by the pulse energy and the ambient temperatThese phonons propagate across the interface and aredetected in the material on the other side. This type ofperiment has been performed for interfaces between soand liquid helium~see, for example, Wichertet al.15! and atsolid-solid interfaces~for a review, see Ref. 16!. Usually, the

FIG. 1. Comparison of selected data for interface thermal conductance~i!individual interfaces measured by picosecond thermoreflectance~Ref. 10!,Al/Al 2O3 ~open diamonds! and Pb/c-C ~filled triangles!; ~ii ! conductance ofthe a-GeSbTe2.5/ZnS interface~open triangles! from a multilayer sample~Ref. 14!, ~iii ! series conductance of the top and bottom interfaces of meSiO2–silicon structures~filled circles labeled by MOS! ~Refs. 12 and 13!.The solid line is the calculated diffuse mismatch conductance of Al/Al2O3

using the Debye model; the dashed line is the theoretical predictionAl/Al 2O3 using a lattice-dynamical calculation of a model fcc interfa~Ref. 10!. The right axis gives the thickness of a film withL51 W m21 K21 that has the thermal conductance corresponding to theaxis.


et

elrord

.

enoton-r-e.hen-

ds

experiments are performed at low temperatures so thatphonon mean free path is long; i.e., the phonons travellistically in the two media. As a result, it is possible to stuseparately the transmission of phonons of different polartion and to observe the refraction that occurs whenphonons cross the interface. This refraction, combined wthe effects of phonon focusing lead to an interesting angvariation of the phonon flux emerging from the interface16

As an example, in Fig. 2 we show the phonon flux emergfrom the interface between two twist-bonded Si crystals17

The faces of these crystals both have a~100! orientation, butone crystal is rotated by 40° around the@100# axis relative tothe other.

To study the propagation of coherent phonons acrossinterface, the picosecond ultrasonic technique can be useshort light pulse is absorbed in a material, and becausthermal expansion a short strain pulse~coherent phonon! isgenerated. This strain pulse is detected after it has beenflected or transmitted across the interface to another mateIn this way the transmission of phonons of frequency up

–

or

ftFIG. 2. ~Color! Top: experimentally observed phonon focusing imagetwist-bonded Si with a twist angle of 40 degrees~Ref. 17!. Each of the twobonded crystals are 3 mm thick, the width of the image correspondsmm. Bottom: simulated image. Arrows indicate the crystal orientation retive to the phonon source. Phonon intensities are normalized to the mmum value in each image.


pee

eona

nthis

et

o

aerunanighmenth

gxt

e

m

tiodnlicte

f-

’sinic

cht,fesi-lyvetthu

veouhe

Dboatcsys-

r’s9ntthei-

tialvery

at,bo

mal.hatenan-is-ce.tion,era-re-

ns

be-in

inugh

sityula-

.


about 1 THz can be studied. In one experiment of this tycoherent phonon transmission at a bonded interface betwSiO2 and Si was studied.18,19 It was possible to use thacoustic transmission as a measure of the quality of the band to see how the bonding improved with thermal anneing. In another experiment,20 a graded interface betweefilms of gold and aluminum was studied. The interface widz was;400 Å. For a diffuse interface the phonon transmsion is expected to depend on the value ofkz, wherek is thephonon wave number. Forkz>1, the transmission should blarger because the acoustic properties vary smoothly onscale of the phonon wavelength. The experiment demstrated this effect.

Quantitative measurements with coherent phonons hmany potential applications for the characterization of intfaces. In picosecond ultrasonics, the light creates sopulses, i.e., a continuum of modes for which the relevlength scale is defined by the penetration depth of the land other parameters of the overlayer. Periodic layereddia offer an alternative method for generating coherphonons. Here, phase-matching considerations showlight can couple to modes of wave vectorsq'2pn/d wheren is an integer andd is the period of the structure. Usinfemtosecond laser sources, this approach has been esively applied to GaAs-AlxGa12xAs ~Ref. 21! andInxGa12xN– GaN~Refs. 22 and 23! superlattices to generatcoherent acoustic modes of frequencies up to;1 THz.

C. Molecular-dynamics simulation of heat transportand phonon dynamics

Computational approaches to heat-transfer problespan the range from numerical solutions of Fourier’s law24 tocalculations based on the Boltzmann transport equa~BTE!25,26 to atomic-level simulations. In the Fourier anBTE approaches, the physics of heat transfer and phoscattering are incorporated into the calculations in an expmanner; thus for a reliable calculation, a fairly sophisticaunderstanding of the fundamental phonon processes isquired. While in some cases~e.g., anharmonic phonon efects, isotopic defects, and point defects! understanding iswell-developed, in others~e.g., interfaces! current under-standing is poor. Molecular-dynamics~MD! simulation, bycontrast, merely involves the integration in time of Newtonequations of motion for an ensemble of atoms interactwith each other through a, usually empirical, interatompotential.27,28 Because the formalism of the MD approadoes not require anya priori understanding of heat transporit is ideal for investigating the fundamental heat-transmechanisms themselves. However, MD does have thenificant limitation of being entirely classical, with each vbrational mode equally excited; thus it is only rigorousapplicable to solids above the Debye temperature. Moreosince electrons are not included in an atomistic model, inot possible to simulate electrical conductors orelectron–phonon interactions present in many semicondtors. In this subsection we briefly review recent atomic-lesimulations of thermal transport at interfaces, and pointthe opportunities for gaining important new insights from tsimulation of individual phonon processes.


,en

d,l-

-

hen-

ve-dtt

e-tat

en-

s

n

onitdre-

g

rig-

r,isec-lt

The two most commonly used approaches to the Msimulation of the thermal conductivity are the Green–Kuapproach in which the equilibrium fluctuations in the hecurrent are analyzed, and the ‘‘direct method,’’ which mimiexperiment by imposing a temperature gradient on the stem and determining the thermal conductivity from FourieLaw ~for a recent review of these methods see Ref. 2!.While simple in principle, in practice each has significacomplications associated with system-size effects andrelatively long simulation times required. Moreover, the drect method has the additional complication of a potennonlinear response of the system associated with thelarge temperature gradients imposed~typically in excess of109 K m21). A recent systematic comparison has shown thwhen extrapolated to infinite system size, the Green–Kuand direct methods give consistent values for the therconductivity, albeit each with error bars of 20% or more29

Despite the additional complications, it was concluded tthe direct method is usually preferable, particularly whapplied to interfacial systems, as it has the significant advtage of providing a direct measure of the interfacial restance,RK , from the temperature drop across the interfa~Unfortunately, however, as discussed in the next subsecthe conceptual issues associated with defining the tempture in an inhomogeneous system have yet to be fullysolved.!

There have been relatively few atomic-level simulatioof heat transport at solid–solid interfaces. Pickettet al.30 ex-plored the transmission of phonon energy at an interfacetween two diamond-structured materials differing onlytheir masses. Maitiet al.31 used the direct method to performthe first simulations of thermal transport through graboundaries. Figure 3 shows the temperature profile throthe simulation cell. The thermal conductance,GK , of theinterface can be calculated from the temperature dropDT,determined from the figure, and from the heat-current denQ passing through the system, determined from the simtion: GK5Q/DT. For two different symmetric tilt grainboundaries in Si, they obtainedGK50.8 and GK

50.9 GW m22 K21.

FIG. 3. Thermal profile calculated by MD for aS513 twin boundary in Si,after Ref. 31. The sample is 25 nm along thex axis and contains 840 atomsTotal simulation time is 1 ns.


-

rn

fTois

foeh--a

thmcaunt

pr

tero

m

ngthta

eoofebfo

ionoteurlio

iothaoals

gagatif

ahen

uc-ionictheatslip

ar-fuln.ent

nnd

edteine

r

seis-

ho-of

eseen

s-

ass

theted


Recently, Schellinget al.32 determined the thermal conductance for the~111! S31 twist GB, which has a rather lowenergy, and for the~100! S29 twist GB, which has a rathehigh energy, for the same interatomic description of Si aobtained values of 1.53 and 0.85 GW m22 K21, respectively.In a number of studies of the GBs in Si, Keblinskiet al.33

showed that the~100! S29 GB is, from a structural point oview, representative of the many high-energy GBs in Si.gether with the very similar values for the two tilt GBs, thsuggests thatGK'0.8– 0.9 GW m22 K21 may be taken asrepresentative of most GBs in Si. The much higher valueconductance of the~111! S31 is reasonable given that thGBs on~111! plane have the lowest energy of all the higangle GBs in Si.33 In comparison with the experimental results show in Fig. 1, these values for the conductancerather high. One possible reason for this is that all ofexperimental systems are interfaces between dissimilarterials which, due to the acoustic-impedance mismatch,be expected to have a lower conductance than a grain boary. A second possible reason could be that the simulagrain boundaries are defect-free and atomically flat, thussumably minimizing the thermal resistance.

It is useful to analyze the thermal resistance of an inface in terms of the length of perfect crystal that would pvide an equivalent thermal resistance. This led Nanet al.34 todefine the Kapitza length,l K5k/GK ; typical values for Si(k;100 W m21 K21 and GK'1 GW m22 K21) give l K

;100 nm as the thickness of perfect crystal with the sathermal resistance as the interface.

Figure 3 shows regions of nonlinear temperature chafar from the GB. These nonlinear regions are close toheat source and heat sink in the system and arise frominterface between the regions that are heated and cooledthe rest of the system. The interpretation of these nonlinregions can draw from the results of phonon transptheory,25,26 which has connections to the kinetic theorymolecular transport35,36 and the equation of photon radiativtransfer.37,38 These increased temperature gradients resemthose predicted by the Boltzmann transport equationsgases, photons, and phonons.

Isothermal boundary conditions enforce perfect emissand absorption of phonons from the boundary, i.e., a phoblackbody, and ensure equilibrium conditions for the emitphonons. This has an influence comparable to an ensphonon scattering event at the boundary. Phonons traveaway from the boundary have a distribution characteristicthe boundary temperature, which differs from the distributof phonons traveling in the opposite direction more thandifference found deep within the medium. This results inincreased departure from equilibrium locally within the phnon system and an increased gradient in the directionaveraged phonon energy. This is called the temperaturephenomenon, and is well documented theoretically formolecules, photons, and phonons. It occurs within a lenscale of the boundary comparable with the mean free pThe magnitude of the temperature slip, specifically the dference between the boundary temperature and the extrlation of the linear temperature distribution deep within tmedium, is approximately the product of the heat flux a


d

-

r

reea-nd-

ede-

r--

e

eehend

arrt

ler

nn

dedngf

nen-lylips

thh.-po-

d

the ratio of the mean free path and the bulk thermal condtivity. This argument is based on the Rosseland diffusapproximation.39 The near-boundary predictions of atomistsimulations should be compared with the solutions tophonon transport equation in the vicinity of boundaries, orleast to the closed-form expressions for the temperaturephenomenon.

Although the value for the thermal conductance of a pticular interface in a particular material may be a usequantity to know, it provides no mechanistic informatioThe explicit relationship between the temperature-dependthermal conductance,GK(T), and the phonon transmissiocoefficient at an interface has been given by Young aMaris:40

GK~T!5E dvC~v,T!^vz~v!&^t~v!& ~2!

whereC(v,T) is the specific heat, which can be determinfrom the phonon density of states and the Bose–Einsdistribution function,vz(v) is the component of the phasvelocity of the phonon normal to the interface andt(v) isthe transmission coefficient. The^ & denotes an average oveall bands and over allk vectors~i.e., for both normal andoff-axis incidence!. While the specific heat and phonon phavelocity can be determined quite easily from the phonon dpersion curves of a perfect crystal, determination of the pnon transmission coefficient requires a detailed analysisthe interface.

Young and Maris40 determined^t(v)& for a nearest-neighbor spring model of a fcc lattice with different massand spring constants on the two sides; this has recently bextended to a disordered interface.41 In both studies, it wasfound that^t(v)& decreases almost uniformly with increa

FIG. 4. Interaction of a wavepacket of longitudinal acoustic phonons@v'8.7 THz, kz'0.62(2p/a)] with an ~001! epitaxial interface between twodiamond-structured lattices differing only in the mass of the atoms (m51 on the right of the interface, mass54 on the left of the interface!; 37%of the energy in the incoming acoustic phonon package is transmitted inform of a slow-moving optical phonon, while the remaining 63% is reflecwith no change in frequency or wavelength~Ref. 42!.


leretobeesooto

ertsceFfit

reisr-

ordsa

cem

ani

thdtly

weTemw

caepeas

eme

nsn

nm

ulieesenthe

nd

ng

e?age3.y inki-thentrop

theeical

ed

iblens

asan-

er

,rs

ureynd

veis


ing v. These results are rather different from a naive impmentation of the acoustic mismatch model, which would pdict a weak dependence on frequency up to a cufrequency determined by the maximum frequency availato the softer material. A simple analytically tractable modthat captures the basic features of the frequency dependin the phonon transmission coefficient would be of great u

The experiments described in Sec. II B point to the ptential capabilities of methods in which phonon packetswell-defined wave vector and polarization are launchedwards individual interfaces. Recently, Schellinget al.42 per-formed atomic-level simulations of phonon–interface intactions. The basic idea is to launch phonon wave packewell-defined polarization and wave vector towards interfaand to analyze the transmitted and reflected phonons.example, Fig. 4 shows four snapshots in the evolution olongitudinal acoustic phonon as it approaches, interacts wand then is partially transmitted through and partiallyflected by an interface.42 Analysis of simulations such as thwill allow the atomic-level details of phonon–interface inteactions to be elucidated.

As discussed above, computational methods basedapplications of Fourier’s Law and the Boltzmann transpequation are well developed. While MD simulation holpromise of providing detailed atomic and spectral informtion on heat flow and phonon dynamics, MD will not replaBTE and Fourier’s-law approaches to heat transfer problein complex materials structures and devices. An importchallenge is therefore the development of techniques tocorporate the insights of atomic-level approaches intohigh-level approaches; success in this can be expecteresult in BTE and Fourier’s Law approaches of significanhigher fidelity.

D. What is temperature?

In the above discussions of atomic-level simulationhave swept an important conceptual issue under the rug.usual definition of temperature is related to the averageergy of a system of particles. This definition is for a systein equilibrium, and works even for nanoscale systems. Hoever, our interest is in the transport of heat through nanossystems. Can temperature, which is an equilibrium concstill be invoked in a nonequilibrium process such as hflow? The answer is affirmative for macroscopic systemthey are so large that one can define a local temperatureach region in space. This local temperature will vary froregion to region. Then one finds, for example, that the hcurrent is proportional to2“T. The question ‘‘What is Tem-perature?’’ is really a question about the size of the regioover which a local temperature can be defined. In masemiconductor superlattices, the layer thickness is 2–5Are these regions large enough to define a temperature?

As discussed above, there have been several molecdynamics simulations of heat flow through grain boundarThe most important conceptual problem raised by thsimulations is how to define the temperature at differplanes in the simulation cell. The simulations calculatepositionRi(tn) and velocityvi(tn) of each atom at each tim


--fflelncee.-f-

-ofsorah,-

ont

-

st

n-eto

hen-

-let,t:in

at

y.

ar-s.ete

step. It is a relatively simple task to store the velocities, acompute an average kinetic energy overN steps in time

^K.E.i&5m

2N (n

N

v2~ tn!5 K 1

2mv i

2L . ~3!

It is found that the averaging must be done over very lotimes to obtain good statistical average kinetic energies.

How is this simple quantity converted to temperaturThe numerical simulations show a sharp drop in the averkinetic energy at the grain boundary, as shown in Fig.Here each point corresponds to the average kinetic energa slice of a few unit-cell thicknesses. The sharp drop innetic energy, at the grain boundary, is consistent withidea of Kapitza resistanceRK which assumes the heat currethrough the boundary is proportional to the temperature dQ5DT/RK . In order to findRK , the kinetic energy must beconverted to a temperature scale.

Because the calculations are entirely classical, withmotion of each silicon atom found from Newton’s law, thmost obvious temperature scale is provided by the classexpression

K 1

2mv i

2L 53

2kBTi . ~4!

The temperature at each atom at siteRi is given byTi . Thisis the procedure adopted in the MD simulations describabove.

However, quantum mechanics provides another possdefinition. The collective excitations of the atomic motioare phonons. They have a frequencyvl(q) which dependsupon the wave vectorq and polarizationl of the phonon.The simulations of Fig. 3 were performed on silicon that hsix polarizations for each wave vector. In quantum mechics the average kinetic energy of the phonons is

K 1

2mv i

2L 51

4N (l,q

\vl~q!j ( i )2~l,q!

3S 2

e\vl(q)/kBTi2111D , ~5!

whereN is the number of unit cells which is also the numbof q points. The polarization vectors arej ( i )2. They are nor-malized so that(lj ( i )253. In the limit of high temperaturewhere kBT@\vl(q) the Bose–Einstein occupation factoare approximated as

2

e\vl(q)/kBTi2111'2

kBTi

\vl~q!. ~6!

In this high temperature limit, the quantum relation Eq.~5!becomes identical to the classical Eq.~4!. The problem withmany simulations is that they are not in the high temperatlimit. Silicon has optical phonons of very high energ(62 meV'750 K). For any temperature between 300 a1000 K, the two definitions of temperature@Eqs.~4! and~5!#gave very different values forTi .

Another possible definition of temperature is to remothe zero point motion from the quantum definition, since it


ibni

chel

a

angininth

scon

ib

ledte

metioe

suif-ro

e.inmfin

iot

aa

oipee

tenehea

g

enoly

ne

m.thethew

ra-

illandondflowho-troleheataveu-

fer-med

und-withs atem-tep

ture.ex-twoain

z--thehisde-uc-rity

nsfor

ise ofareedx-

mp-

de-the, innethele.vence


not a classical effect. This option now makes three possdefinitions of local temperature. Which is the correct defition?

An important issue is the size of the region over whitemperature is defined. The classical definition is entirlocal, and one can define a temperature for each atomplane of atoms. For the quantum definition, the length scis defined by the mean-free-path,lq of the phonon. If tworegions of space have a different temperature, then they ha different distribution of phonons. The phonons can chatheir distribution by scattering. The most important scatteris the anharmonic process in which one phonon dividestwo, or two combine to one. This process occurs onlength scale of the mean free path. A local region withdesignated temperature must be larger than the phonontering distance. However low frequency phonons have a l,lq , and high frequency phonons have a short,lq . For thephonons which carry most of the heat, one can plausdefine an average mean-free-path: This,lq is typically largerthan the length of the MD simulation cell: this case is calthe Casimir43 limit. One can scale the results to an infinisized system using the method of Oligschleger and Scho¨n.44

This phonon viewpoint of temperature implies that teperature cannot be defined for a particular atom, or a planatoms. In particular, there should not be an abrupt variain temperature between a plane of atoms. Although this dnition seems quite reasonable, it makes the numerical reshown in Fig. 3 quite puzzling. The MD simulations by dferent groups do show an abrupt change in the kinetic eneof a plane of atoms at the twin boundary. Regardlesswhich temperature scale is adopted@Eqs.~4! or ~5!#, a graphof temperature versus distance will show an abrupt changpossible resolution for this puzzle is that a grain or twboundary may form a natural boundary for a region of teperature. The statement that temperature cannot be dewithin a scale of distance given by,lq may not apply acrossgrain boundaries. The boundaries may provide natural limto the regions of temperature. Even if one adopts this hypesis, it still means that temperature cannot vary withingrain, or within a superlattice layer, on a scale smaller th,lq . If the layer thickness of the superlattice is less th,lq , then one cannot defineT(z) within this layer. Thewhole layer is probably at the same temperature. This pis emphasized, since all theories of heat transport in sulattices have assumed that one could define a local tempture T(z) within each layer.

The results in Fig. 3 may be measurable. X-ray scating at synchrotron sources can measure the lattice constaatoms in each atomic layer.45 The lattice constant can bconverted to a local temperature scale for that plane. Tshould be able to determine whether this quantity hasabrupt change at an interface while heat is flowing. Althouthis discussion of ‘‘What is Temperature?’’ has been cast inthe framework of MD simulations, the issues are more geral. One of the major issues of thermal transport in nancale systems is whether temperature can be defined localit cannot, then how is transport calculated?

Another approach is to adopt a Landauer formalism, aassume there are two thermal reservoirs at known temp


le-

yorle

vee

gtoeaat-g

ly

-ofnfi-lts

gyf

A

-ed

tsh-ann

ntr-

ra-

r-t of

yn

h

-s-. If

dra-

tures, and consider the ballistic flow of heat between theThis does not work for most nanoscale devices sincevalues of,lq are much smaller than the distance betweenreservoirs. So heat flow is diffusive rather than ballistic. Hodo you calculate diffusive heat flow without a local tempeture?

As an example of the importance of temperature, we wcomment on two related theoretical calculations. StonerMaris10 measured the Kapitza resistance between diamand other crystals. Between diamond and Pb the heatwas much higher than could be explained by a purely pnon conduction. Huberman and Overhauser46 suggested thathe conduction electrons in Pb were playing an essentialin the boundary resistance. Phonons from diamond carryto the interface, and electrons in Pb carry it away. There hbeen two different calculations of this process, one by Hberman and Overhauser,46 and one by Sergeev.47 The twotheories are very different since the two groups made difent assumptions regarding temperature. Sergeev assuthat there was an abrupt change in temperature at the boary between diamond and Pb. His assumption agreesthe conventional view of Kapitza resistance which assumetemperature step. Huberman and Overhauser assumedperature could not be defined locally, and there was no sin temperature. The interface region was at one temperaThey used a Landauer formalism to compute the energychange between electrons and phonons. Which of thesedifferent theories is better depends on whether a grboundary can support an abrupt change in temperature.

E. Transport theory

Most theories of transport in solids employ the Boltmann transport equation~BTE!. For both electron and phonon transport, the form of the equation, and the form ofvarious scattering mechanisms, are very well known. Ttheory can explain, in bulk homogeneous materials, thependence of the electrical conductivity, the thermal condtivity, and the Seebeck coefficient, on temperature, impucontent, isotope scattering, and quantum confinement.

However, the Boltzmann equation treats the electroand phonons as classical particles. One is solving, sayphonons, for the densityf @r ,vl(q),t# of excitations withpolarizationl, wave vectorq, frequencyvl(q) at pointr attime t. The wave nature of the excitation is neglected, asany interference phenomena caused by the wave naturthe phonons. Furthermore, the scattering rates in the BTEcalculated assuming the system is only slightly perturbfrom equilibrium. The solutions to the BTE assume the eistence of a local temperature. Neither of these two assutions may be valid in nanoscale devices.

Wave interference becomes important in nanoscalevices. The wavelengths of the phonons are similar tolength scale of the microstructure. At room temperaturemost solids, all of the phonons states in the Brillouin zoare involved in the transport. Their wavelengths spanrange from atomic dimensions to the size of the sampMost of the actual heat is carried by the phonons with walengths of a few nanometers. On this scale wave interfere


padm

thF

suitoatthlsopeBTulo

nn

thow

emachtr

syetho

ayi

oo

ne

sualdi,onrrafefre

isja--

s,tivetal

nd-yersnsohe-

essa

ntter-ili-ure

reinsiontra-n

resrs,er-e-ing.g

gataainples


becomes important in transport. Similar considerations apto electron transport in nanoscale devices. Since the trtional Boltzmann equation cannot deal with these phenoena, other theoretical approaches are required.

One method is to calculate the phonon states foractual microstructure, and use those states in the BTE.transport in a superlattice, the frequenciesvl(q) are those ofthe phonons in the superlattice. This approach insertssome of the wave phonmena. The phonon states dependnificantly on the interference due to scattering from the mtiple layer boundaries of the superlattice. One problem wthis approach is that it ignores an important phenomenMost scattering effects contribute to a finite mean-free-pwhich has a major effect on the phonon frequencies insuperlattice. Solving this problem using the BTE is acomplicated by phonon band folding. There are many sulattice bands, phonons scatter between them, and thebecomes a matrix equation of large dimension. This calction is quite ambitious and has never been done. This dnot include all wave interference phenomena.

Another possibility is to use the quantum Boltzmaequation48 to solve for the distributionf (R,t:v,q). Com-pared to the BTE, there is one more vector variable inargument that makes the solutions more complex. This mfundamental equation does include wave information. Hoever, it is difficult to solve and is seldom used. One problis the choice of boundary conditions to apply on the surfof the system in order to connect it to a heat reservoir. Tissue has been much discussed in the literature on elecflow in nanoscale devices.49,50

If one throws out the concept of a local temperature,that the BTE cannot be used, then there is no known wasolve this important problem. Many semiconductor devicare dependent upon the electrical currents provided byelectrons. The heat currents have components from belectrons and phonons. Presently there is no accurate wmodel the heat flows in these nanoscale systems whilecluding the exchange of heat between electrons and phonA major theoretical challenge is to invent a new methodsolving this and related transport problems.

III. PHONONS AND THERMAL TRANSPORT INNANOSCALE STRUCTURES AND DEVICES

A. Silicon films and devices

1. Conduction along semiconducting monolayers

Phonon conduction along single- and polycrystallimonolayers is important for silicon-on-insulator~SOI!circuits,51 novel 3D multilayer electronics,52 and MEMSwith semiconducting or dielectric membranes.53 Siliconmonolayers have received the most attention, either aspended membrane~a geometry relevant for many thermmicrosensors! or embedded between amorphous siliconoxide films ~as in SOI transistors!. In the second geometrythe surrounding silicon dioxide films act as diffuse phonemitter absorbers and provide a boundary condition thatsembles phonon scattering on the boundaries of a membwith roughness above a few angstroms. This situation diffrom that in superlattices with comparable phonon mean


lyi--

eor

inig-

l-hn.he

r-E

a-es

ere-

eison

otosethto

n-ns.f

s-

-

e-ne

rse

path on both sides of each interface, for which conductioninfluenced by the coupled ballistic phonon transport in adcent layers, see Sec. IV.~A macroscopic version of this geometry has been used by Pohl and co-workers54 to studyphonon scattering in thin films atT,1 K.)

Figure 5 shows that for single-crystal silicon filmphonon-interface scattering strongly reduces the effecthermal conductivity at low temperatures. For single-cryssilicon layers doped at concentrations higher than 1017 cm23

the conductivity is reduced by scattering on impurities afree electrons.55 Figure 5 shows that this is particularly important at low temperatures. Measurements at 20 K on laof thickness 3mm with phosphorus and boron concentratioof 1.031018 cm23 yielded reductions by approximately twand four, respectively, with the difference resulting from tdisparity in mass of the two impurity types. At room temperature, measurements along silicon films of thickndown to 74 nm showed a reduction of the conductivity byfactor of two compared to bulk data.

In polycrystalline silicon films, phonon scattering ograin boundaries and related defects dominates over scaing from the surface or the film/substrate interface. Polyscon conductivities are reduced compared to those of pcrystalline films,56 but the roles of impurities and grains adifficult to separate. Recent work studied films with grasizes between 300 and 500 nm, extracted using transmiselectron microscopy, and boron or phosphorus concentions up to 4.131019 cm23, measured using secondary iomass spectroscopy~SIMS!.57,58 Figure 5 shows that polysili-con conductivities are reduced strongly at all temperatucompared to similarly doped single-crystal silicon layewhich illustrates the importance of grain boundary scatting. The grain structure and spatial impurity distribution dpend strongly on the temperature and duration of annealAnnealing of the doped polysilicon film and the resultin

FIG. 5. Overview of thermal conductivity data for silicon films with varyinmicrostructural quality and impurity concentrations after Ref. 58. The dillustrate the relative importance of phonon scattering on film and grboundaries and impurities. The data for undoped single-crystal bulk samare taken from Ref. 59. Both single crystal films have thickness of 3mm,and for the doped film, a boron concentration of 1019 cm23. ~Ref. 55!. Theundoped and doped (1.631019 cm23 boron! polycrystalline films have grainsizes of 200 and 350 nm, respectively~Refs. 57 and 58!.


it

lyetionex

feree

aiaheuc

omstl

ionAsgmffedniciceaf

ofalioth

ionons,theortal-itsl-forlar

ethe-

ver-uchblyed,abe-andlexringe-

ns-stichasonteleswn

byeer

andust inivi-ingra-edge.othhisyingher-

rensan-onsare

ort.dnsui-it isan-

o

atons


increase in grain size yields a higher thermal conductivthan that of the undoped film.

The conduction size effect along crystalline and pocrystalline monolayers at low temperatures can be effectivpredicted using a semiclassical approach, based on soluto the Boltzmann transport equation for conduction alofilms60 and based on a spectrally-resolved form of thepressionk5Cv,/3.61 In this expression,C is the heat ca-pacity per unit volume andv and, are the phonon velocityand mean free path, respectively. Polysilicon data are eftively predicted below about 10 K by reducing the mean fpath using Matthiessen’s rule62 and a grain-boundary frepath given bydG /B, wheredG is the grain size andB is adimensionless constant governed by the grain shapegrain boundary reflection coefficient. Polycrystalline dmond films are worthy of brief discussion here because tillustrate another way in which grain boundaries can redphonon conduction. The free pathdG /B is overly simplisticfor diamond film data owing to the large concentrationother imperfections. Because imperfections populate pririly at grain boundaries, the phonon scattering rate canbe coupled to the grain sizedG using a semiclassicaapproach.63

The anisotropy and nonlinearity of phonon dispersrelationships complicate predictions above 100 K for GaGe, and AlAs films, and above 150 K for silicon films owinto its higher Debye temperature. The predictions becomore challenging because phonon dispersion induces diences in transport among the various phonon modes anrections. Figure 6 illustrates this issue for transport alosilicon films of thickness near and below 100 nm. Predtions are performed under a greybody assumption, whoverestimates the fraction of phonons contributing to htransfer and therefore underestimates the phonon meanpath and the thermal conductivity size effect for a filmgiven thickness. Predictions are also provided with spectrvarying mean free paths under the competing assumptabout which phonon modes are dominating conduction inmaterial.

FIG. 6. Experimental data for the effective room-temperature thermal cductivity along silicon films of thickness near 100 nm~Ref. 64!. The dataare compared with predictions based on the Boltzmann transport equengaging different assumptions about the frequency dependence of phrelaxation times and the relative contributions of the phonon brancheconduction.


y

-lyns

g-

c-e

nd-ye

fa-ill

,

er-

di-g-ht

ree

lynse

2. Monte Carlo solutions of the Boltzmann transportequation

The problem of phonon transport considering disperseffects has been addressed using Monte Carlo simulatiwhich are essentially an alternative approach to solvingBoltzmann transport equation. The Boltzmann transpequation~BTE! describes heat flow by phonons in nonmetlic solids, when wave effects are negligible. The BTE, inmost general form, is difficult to solve analytically for reaistic phonon dispersion and density of states as well astransitions between phonon polarizations and for irregugeometries.65 Recent work-by Majumdar,26 Chen and Tien,66

Goodson,63 and Chen67 have used analogies from radiativheat transfer and presented solution strategies for bothdiffusion and the ballistic limit. In such calculations, however, phonon dispersion was neglected, and a single ‘‘aage’’ polarization branch was considered. Furthermore, scalculation strategies can only be adopted for considerasimplified geometries. If all these assumptions are removit is extremely difficult to solve the BTE for phonons bydeterministic approach in an arbitrary geometry. This iscause the number of independent variables is too large,would render any kind of discretization scheme too compto be practical. Furthermore, the phonon–phonon scatteevents are difficult to incorporate without a simplifying rlaxation time approximation.

One approach to capture the complexity of phonon traport in the particle regime is to use Monte Carlo or stochasimulations to solve the BTE. Over the last decade therebeen tremendous advancement in the development of MCarlo solution techniques for the BTE for electrons and hoin semiconductors.68–72 However, there have been very fereports of using Monte Carlo simulations for phonotransport.73,74 These studies have either assumed the Deapproximation for density of states or did not considphonon–phonon interactions. More recently, MazumderMajumdar75 considered phonon dispersion as well as variophonon scattering mechanisms to study heat transporcomplex geometries and to predict the thermal conductties. The result showed that by fitting one parameter usexperimental data of thermal conductivity at one tempeture, predictions of the thermal conductivity of silicon agrewell with experimental data over a wide temperature ranIn addition, it was able to capture phonon transport in bthe ballistic and diffusive limits. Despite the success of tapproach, one must be aware of the challenges in applthis approach more generally to problems in nanoscale tmal transport.

First, it must be recognized that it is difficult to captuall the selection rules for transitions between polarizatioduring phonon–phonon interactions. For example, the trsitions between longitudinal and transverse acoustic phonas well as those between acoustic and optical phononsvery important in energy dissipation and thermal transpMazumder and Majumdar75 ignored the selection rules anobtained local phonon distributions in different polarizatiobased on a quasiequilibrium condition. For highly noneqlibrium cases, this approach cannot be applied. Hence,extremely important to develop schemes for capturing tr

n-

ionon

to


tei

lidobfitlamucoveh

isisuhenveno

fae

leoncdec-reeclevea

alsndo

aicfu

ireno

.ann

oditf

th

alto

h in-

x-eti-ta-

theonssta-lec-

in-porttorticethe

onartla-s,

allyadic-

. 7.ere

h attheer, theo-

us-ndthesion

tive

d ond afree


sitions between polarizations during phonon–phonon inactions. Second, the relaxation times for phonon–phononteractions have, in the past, been used only for bulk soand only for the most common materials. These weretained from approximate analytical solutions that requireting parameters. For example, the phonon scattering reation times for silicon are generally obtained froHolland.61,76These cannot be used when silicon is nanostrtured which can modify the phonon dispersion relation. Fmany other materials, relaxation times are not available efor bulk solids. Hence, there is a need to develop approacof predicting the relaxation times for arbitrary phonon dpersion relations. This would allow one to study energy dsipation and thermal transport in nanostructured solids sas quantum dots, nanowires, films, and superlattices, wthe phonon dispersion relations are modified due to confiment or wave interference. If both these challenges are ocome, Monte Carlo simulations of phonon transitions atransport can be an extremely effective tool to study nancale energy dissipation and thermal transport.

3. Nanoscale hotspots in devices

The phonon-boundary scattering discussed abovesilicon monolayers is only the first among three nanosceffects which complicate heat conduction from metal-oxidsemiconductor~MOS! transistors. Two additional nanoscaeffects result from the small size of the region of electrenergy transfer to the lattice. Before describing these effeit is important to note that all three are augmenting theparture from equilibrium of the phonon distribution funtions, rendering the usefulness of a phonon ‘‘temperatumore questionable. Theoretical treatments of these effuse temperatures either for the phonon system as a whofor fractions of the phonons sharing comparable grouplocities; this approach can at best be viewed as a shorthplaceholder for energy density. This situation may eventube remedied through detailed phonon transport analysimolecular dynamics, although the computational demawill be excessive owing to the three-dimensional naturethe problem.

Electrons traveling between the source and drain arecelerated by the electric fields and generate primarily optphonons. Simulations of electron transport using drift difsion and the more rigorous Monte Carlo or hydrodynamapproaches77,78 illustrate that rates of phonon generation asharply peaked on the drain side. The region with strolattice heat generation has dimensions near tens of naneters and is much smaller than the channel length~50–150nm depending on the technology generation of interest! andthe phonon mean free path,,'250 nm according to Ref. 64This second nanoscale effect, which we will call hotspot blistic emission effect, causes the temperature of the pho‘‘hotspot’’ to be larger than that predicted using diffusiotheory, by a factor comparable with,/r where r is thehotspot dimension.79 While the precise distribution of thegenerated phonons between the optical and acoustic mdepends strongly on the energies of scattering electrons,clear that the generated phonons have group velocitiesbelow the velocity characteristic of heat conduction by


r-n-s-

-x-

-rn

es--chree-r-

ds-

orle-

ts,-

’’tsor-

ndlyorsf

c-al-c

gm-

l-on

esisare

lattice. The generation of high concentrations of opticphonons within a region with dimensions small comparedthe mean free path causes a third nanoscale effect, whicthe present manuscript we will call the hotspot far-fromequilibrium effect.

Sverdrup80,81 examined this third nanoscale effect eperimentally using a dedicated microstructure and theorcally by breaking the phonons into two propagating and stionary groups. The theoretical treatment solvedBoltzmann transport equation for the propagating phontogether with a reservoir energy-storage equation for thetionary phonons. Stationary phonons were generated by etron scattering, with a spatial distribution taken from andependent analysis of the electron and hole transproblem in the transistor. Simulations of a SOI transiswith channel length 150 nm concluded that the peak lattemperature~a weighted average of the temperatures ofstationary and propagating phonons! was elevated by morethan a factor of two compared to predictions based onlythe heat diffusion equation. This difference resulted in pfrom the localized transfer of energy from phonons with retively small group velocities to those at lower energiewhich dominate heat conduction.

This third nanoscale effect was examined experimentusing localized heating by a doped silicon bridge withinsuspended silicon membrane. Experimental data and pretions for the temperature of the bridge are provided in FigThe cross-sectional dimensions of the doped heater w'1 mm and much larger than the phonon mean free patroom temperature, such that all of the predictions anddata are consistent with simple diffusion theory. At lowtemperatures, where the phonon mean free path is longerdramatic departure from equilibrium induced by optical phnon generation within the hotspot~far-from-equilibrium ef-fect! becomes important. Predictions at low temperaturesing the Boltzmann equation with the stationary apropagating phonon groups agree reasonably well withdata, and are consistent with a simple closed-form expresin which the excess temperature rise scales with (,/r )2.

FIG. 7. Experimental data for the phonon temperature rise within a resisheater in silicon with dimensions~at low temperatures! smaller than thephonon mean free path. The data are compared with predictions basediffusion theory, the two-fluid phonon Boltzmann transport equation, ansimple resistance model based on the ratio of the square of the meanpath and the effective cross-sectional area of the heat source~Ref. 81!.


ulusuthveonissthhe-th

uoccen

abo

in

nfrethdihet

anh

est

onds

rinhrar

on

rcmfo

bco

allyt.

es.tbebe

it ision

centc-eakrmalat

andr andand

bri-ts ofr-

eaterualds


The hotspot effect, discussed above for silicon, covery well be much larger in optoelectronic devices becaof the phonon band structure in the associated semiconding materials. In polar semiconductors, such as GaAs,carriers interact predominantly with LO phonons of wavector q'0.82 Thus, carrier cooling leads to the generatiof nonequilibrium modes for which the group velocityexceedingly small, i.e., to a nanoscale hot spot. Thiscalled hot-phonon effect is believed to be responsible forreduction of the cooling rate of electrons at densities higthan 1017 cm23.83 It is well known that the presence of interfaces changes the character of the optical phonons inthey are no longer purely LO or transverse-optical~TO! but amixture involving LO, TO, and interface modes.84 Sincemixing results in larger dispersion, this suggests that strture engineering can be applied to attain large group velties to facilitate the removal of LO heat from the deviactive region. In the bulk, heat diffusion relies on the spotaneous decay of the optical phonons into the more movacoustic phonons.85 Recent progress in the generation of cherent high-frequency acoustic modes,86 which can be usedto stimulate LO decay, hold promise for applicationsnanoscale heat removal.

B. Nanostructures with tailored phonon transportproperties

Unlike bulk materials, the phonon properties of semicoductor nanostructures and, in particular, the phononquency, group velocity, spectral density as well asstrength of the interaction with carriers, can be widely mofied to improve the performance of optoelectronic and otdevices. In some cases, it is desirable to increaseelectron–phonon interaction; an example is the interbsemiconductor laser, where fast relaxation rates reduceelectron effects.87 On the opposite side, we have devicsuch as mid-infrared emitters and detectors based on insubband transitions, for which a reduction of the electrphonon coupling constant is advantageous because it lealarger lifetimes of the relevant excited states.87 An importantexample of the effect of quantum confinement on scatterates is that of the phonon bottleneck in quantum dots. Trefers to the strong suppression of the electron relaxationwhich arises when the energy spacing of the confined caers exceeds the energy of the longitudinal optical~LO!phonon.88

C. Phonon transport in 1D nanostructures

One-dimensional~1D! nanostructures such as carbnanotubes and semiconductor nanowires have recentlyceived a lot of attention. While most of the current reseais being focused on electronic and optical properties, thertransport is also of interest for basic science as well astechnological applications. For example, Schwabet al.89

demonstrated quantum thermal conductance in nanofacated 1D nanostructures, where the quantum of phononductance,Gph , is given


dect-e

o-er

at

c-i-

-le

-

--

e-r

hedot

er--to

giste

ri-

re-halr

ri-n-

Gph5p2kB

2T

3h, ~7!

wherekB is the Boltzmann constant,T is the absolute tem-perature, andh is the Planck constant.

In this regime, a 1D nanostructure behaves essentilike a phonon waveguide similar to optical ones for lighBerber90 and Osman and Srivastava91 have theoretically pre-dicted very high thermal conductivity for carbon nanotubIn addition, Hicks and Dresselhaus92 have suggested thaquantum confinement of electrons in nanowires couldused to manipulate thermoelectric properties, which coulduseful for solid-state energy conversion devices. Hence,important to study thermal transport and energy conversin 1D nanostructures.

On the experimental side, there have been some reattempts93 to measure the thermal conductivity of a colletion or a mat of carbon nanotubes. However, due to wcoupling between carbon nanotubes, the measured theconductivity was found to be much lower than thpredicted.90 Kim94 developed a microdevice~see Fig. 8! con-taining two adjacent silicon nitride (SiNx) membranes or is-lands suspended with long thermal isolation legs. Each islcontains a Pt resistor which can be used as both a heatea thermometer. A nanowire can be placed in between

FIG. 8. A large scale scanning electron microscopy image of a microfacated device~left!. Two independent islands are suspended by three se250 mm long silicon nitride legs with Pt lines that connect the microthemometer on the islands to the bonding pads,~right! Enlarged image ofanother design of suspended islands with the Pt thermometer and hfabricated by electron beam lithography. Also shown is an individmultiwall carbon nanotube, 14 nm in diameter, bridging the two islan~Ref. 94!.


sr-ioetr

codu

a

2uteaanitehcao

g

oirosin

cada

on-difych.

twongeandec-ires

duc-nsareles?hat

onfectsde-n-y and

oratan-isor-dataiusar

edass

gare

edheofer

in

ar

de

ce

andver


bridge the two suspended islands. One of the islandheated to a temperatureTh , which is determined by measuing the resistance of the Pt wire on the island. Conductthrough the nanowire heats the sensing island to a tempture Ts , which is also measured by resistance thermomeBy noting the power dissipated by the heater,Qh , and esti-mating the thermal conductance of the suspended legsnected the islands in the absence of a nanowire, the contance of the nanowireGn can be calculated.

Figure 9 shows the plot of thermal conductivity ofmultiwall carbon nanotube~MWCN! as a function of tem-perature measured by this technique. TheT2 temperature de-pendence suggests that MWCN behaves thermally as asolid; the MWCN has a thermal conductivity of abo3000 W m21 K21 at room temperature, which is in closagreement with the predictions of Osman and Srivastav91

Note that the measured value includes the contact resistbetween the nanotube and the two islands, althoughpredicted to be lower than the resistance of the tube itsEstimation of the contact resistance has remained a clenge. Nevertheless, it is envisioned that this techniquebe used to measure thermal conductivities of a varietysemiconducting or metallic nanowires that are technolocally important.

Phonon transport in 1D nanostructures offers the opptunity to understand the basic science of phonon dynamand transport, while allowing the ability to manipulate themal properties. Phonons in nanowires can be different frthose in bulk semiconductors mainly because the disperrelation could be significantly modified due to confinemein two directions. In addition, the presence of a surfaceintroduce surface phonon modes. These result in manyferent phonon polarizations other than the two transverse

FIG. 9. Measured thermal conductivity of a 14 nm diameter multiwall cbon nanotube~MWCN! ~solid circle!, an 80 nm diameter MWCN bundle~solid triangle!, and a 200 nm diameter MWCN bundle~solid square! ~Ref.94!. Data for two vapor-grown graphite fibers~Ref. 95!, one heat treated to3000 °C ~open triangle! and one without heat treatment~open circle! areincluded for comparison. The lines represent the calculated~Ref. 95! basal-plane thermal conductivity of graphite, assuming temperature-indepenlow-temperature phonon mean free path,52.9 mm ~upper line! and ,53.9 nm~lower line!.


is

nra-y.

n-c-

D

.ceislf.al-nf

i-

r-cs-montnif-nd

one longitudinal acoustic branches found in bulk semicductors. Such changes in the dispersion relation can mothe group velocity and the density of states of each branThe phonon lifetime also changes and this arises fromsources. First, the phonon–phonon interactions can chabecause selection rules based on energy conservationwave-vector relations depend on the dispersion relation. Sond, boundary scattering can be much stronger in nanow~5–50 nm diameter! than in bulk semiconductors. Whilethere have been some theoretical studies on phonon contion in nanowires,96,97 there are still several open questioregarding phonon transport in 1D nanostructures: Whatthe phonon–phonon relaxation times and the selection ruHow does the phonon gas couple to the electronic one? Wis the role of individual defects in a nanowire on phontransport and does the presence or absence of single dealter overall thermal transport? How can nanowires besigned for very high or low thermal conductivities? It is evisioned that these questions can only be answered bcombination of experimental studies of single nanowires atheoretical and computational~molecular dynamics andMonte Carlo simulation! studies of phonon transport in 1Dnanostructures.

D. Nanoscale 3D conduction and mass transport inpolymers

Section III focused on heat transport in crystallinenearly crystalline materials. Recent developments in dstorage technology highlight important shortfalls in our uderstanding of nanoscale heat and mass transport in ddered materials such as polymers. In thermomechanicalstorage, a heated AFM cantilever forms a divot with radof curvature below 20 nm in a polymer film of thickness ne30 nm on a silicon substrate.53 Figure 10 illustrates divotswritten in a polymer film, as observed using an AFM-basthermal readback mechanism. The transport of heat and min the vicinity of the cantilever tip governs the bit writinrate and bit size; therefore, nanoscale transport propertiesthe critical figures of merit. Fundamental study is motivatby the fact that the thickness of the polymer film and tradius of curvature of the divot are more than an ordermagnitude smaller than the radius of gyration of the polymin bulk form. Therefore, heat flow and deformation with

-

nt

FIG. 10. Images of divots written in a polymer film using heat and forapplied by a silicon atomic-force-microscope~AFM! cantilever. The imagesare recorded using the same cantilever used during the writing processa thermal reading technique involving heat conduction from the cantileinto the substrate~Ref. 53!.


n-

chfve

ti-drst

apetola,

is

lsinnotahilatmteilcaax

t ifere

nounelng

urizeru

ugintrhgtoga

emto

en

theort.

on-ar-

ertinger-ck-tedlas-leceshighHy-ionFur-num-veblyill

nte-into

isedher-m-rs.-r

oc-on-di-.25ys

nalandte-er

ataro-

ialsh isv-e

em-mi-

tesonal

er-r of


the polymer will be strongly influenced by the partial orietation of molecular strands in the plane of the film.

While the theory of polymer motion has evolved muin the past two decades,98,99 there is little theory and data odirect relevance to the three-dimensional motion obserduring bit formation. The three-dimensional, multiscale~10–100 nm! nature of the conduction problem within the canlever and into the polymer film may well render it beyonthe reach of atomistic simulations for a number of yeaFurthermore, semiclassical phonon simulations such asBoltzmann equation and Monte Carlo techniques, whilepropriate in the silicon cantilever tip, are of little use in thdisordered polymer. Modifications of continuum theoryaccount for anisotropy and partial orientation of molecustrands provide some help with the heat transport problem100

and may become more accurate when coupled with atomsimulations of individual molecules.

IV. THERMAL TRANSPORT IN NANOSTRUCTUREDMATERIALS

In this section we consider bulk or thin film materiathat contain a collection of nanostructured units. Thiscludes amorphous or glassy materials, materials with nacale grains, as well as nanoporous materials that convoids on the order of a few nanometers. Also included in tcategory are multilayer films as well as epitaxial supertices, which include a collection of nanometer-scale filstacked on each other. The common feature in these maals is that one can identify a nanostructured unit as the buing block. For example, in amorphous materials the loordering is on the scale of lattice constants while for epitial superlattices the unit is a superlattice period.

While phonon dynamics and transport in a single uniimportant and must be studied, the collective behavior olarge number of units can add to more complexity, i.e., thmay be some collective modes which are not found whconsidering only single units. For example, when the phocoherence length scales are larger than the size of thephonon interference effects lead to modified dispersion rtions. Hence, the appearance of both wave effects at nacales and diffusive heat flow at bulk scales poses challenin predicting thermal transport in these materials.

Such predictions are important because nanostructmaterials find many applications. For example, with the sof electronic devices shrinking it is necessary to use matals with low dielectric constants. Amorphous or nanoporomaterials are generally chosen for this purpose, althotheir thermal conductivities are very low which resultspoor thermal management. Can materials with low dielecconstants and high thermal conductivity be designed? Tcan be answered if we have a fundamental understandinphonon dynamics and transport in such materials. Refracceramic films are also used in thermal barrier coatings inturbine engines. The performance of such engines cansignificantly improved if the thermal conductivity of thesfilms can be reduced, thus allowing operation at higher teperatures. Finally, epitaxial superlattices of semiconducfilms are under intense investigation for thermoelectric


d

.he-

r

tic

-s-ins-sri-

d-l-

saennit,

a-os-es

edei-sh

icisofrys

be

-r-

ergy conversion devices because they offer the ability tocontrol both electronic band structure and phonon transp

A. Amorphous and nanoporous materials

Heat transport is an important problem to the semicductor industry, which is manifested in at least two keyeas: advanced packaging and device reliability. In the formcase, the challenge is how to maintain acceptable operatemperatures with an increasingly dense circuit layout opating at increasingly higher frequencies. In short, the paage cooling requirement for silicon logic devices is projecto be greater than 175 W after 2010.101 The package thermaproblem is amplified when one considers that the increingly widespread proliferation of logic chips requires flexibpackage platforms for applications ranging from applianand personal digital assistants at one extreme to thecompute node density required in modern server farms.brid, chip-on-chip, and three dimensional chip constructadds to the package complexity that must be addressed.thermore, as performance requirements increase, as theber of I/O’s multiplies, and as the clock frequencies moabove 10 GHz, a shift toward optical interconnects probabecomes inevitable. If so, an additional thermal problem wbe realized because optical transmitters will need to be igrated into the package or perhaps on-chip. Researchnew ways to improve the package thermal performanceongoing, much of which relies on novel opportunities offerby nanostructured materials. Examples range from high tmal conductivity package fill or epoxy using nanotube coposites to chip-scale integrated thermoelectric refrigerato

The device reliability thermal problem refers most importantly to the reduction in electromigration lifetime fometallic interconnect systems, in which metal mass flowcurs in response to electrical current carried in the intercnects until the metal line fails with a high resistance contion. CMOS technology at dimensional scales below 0mm has significant performance limitations due to RC delain the interconnect network, so much so that the traditioAl based metallization has been discarded in favor of CuCu alloys and the silicon dioxide interlayer dielectric marials are being replaced by more exotic materials with lowdielectric permittivity. In the latter case, SiO2 thin films witha relative permittivitye r'4 is forecast to be replaced instepwise fashion by materials withe r'3 and soon after thaby materials withe r'2. This transition is being made atrate limited by our understanding and control of the micscopic and processing properties of the films.

The advantage offered by these new insulator materhas a concomitant set of problems, not the least of whican amplification of the self-heating problem. Low permittiity materials generally have low thermal conductivity, thconsequence of which is an increase in the operating tperature of the metallic traces, leading to reduced electrogration resistance and premature circuit failure. Shen102 pub-lished a straightforward calculation that clearly demonstrathis self-heating issue. Examining a multilevel metallizatiidealization employing Cu and polyimide, as a prototypiclow permittivity material, the temperature increase in intconnect system was calculated as a function of the numbe


peasere

cate

he

th

Fsa

osi

ael

-vi-

S

ities,

stiv-ro-

op-, an

beit

dmale itve-

e-ingnlyofseicaling

in

ge-ltantrob-ry

ials,re-ityneher-tiontoin-therel-on-thepo-dsethe

ort

n–

-s

or

icoce


metal layers in the design. For only a six layer system, tycal of today’s logic designs, the increase of temperatureily exceeded self-heat temperature limits currently in uFuture designs will include more metal layers. Furthermogiven that logic circuit design employs a wide range of lometal densities, it is likely that Shen’s model underestimathe reliability jeopardy.

Given the above considerations, understanding the tmal conductivity of thin (;,500 nm) interlayer dielectricsis of considerable interest. Using an example, considerproperties of carbon doped silicon dioxide~CDO!.103A sche-matic representation of these materials is presented in11, in which thesp3 network of the amorphous oxide iopened up by the introduction of chemical moieties, suchthe methyl group suggested in the figure. The thermal cductivity of a set of these materials has been measured uthe time domain thermoreflectance~TDTR! method,104 andthe results in two cases independently validated usingelectrical resistance measurement and full scale finite

FIG. 11. ~Color! Schematic representation of methyl doping of an amphous SiO2 network.

FIG. 12. Room temperature thermal conductivities of carbon-doped sildioxide ~CDO! films measured using time domain thermal reflectanTDTR. ~The linear fit is merely a guide for the eye.!


i-s-.,ls

r-

e

ig.

sn-ng

ne-

ment thermal modeling method.105 The results are summarized in Fig. 12, in which the measured thermal conductities of thermally grown oxide, plasma enhanced TEOoxide, fluorine doped SiO2 , a-SiC, and various forms ofCDO are compared to the measured oxide mass densdetermined using x-ray reflectometry.~The a-SiC samplesare actually a carbon rich CDO material.! The variability ofthis measurement is estimated to be;5% determined bytesting multiple samples and over a period of'1 year. Forthis family of materials the introduction of carbon is the moimportant factor correlated to the drop in thermal conductity. The preparation method and the chemical precursor pduce smaller changes in thermal conductivity. Carbon ding, in turn, reduces the resultant mass density of the filmeffect which is amplified by the inclusion of (CH)x moietiesfor x51, 2, and 3 in the amorphous network.106,107Note alsothat F doping has the same density reducing effect, alsmaller in magnitude.

The minimum thermal conductivity theory of Cahill anPohl108 has been applied to these results to derive therconductivities. This model is an attractive option becausdepends only on the atomic number density and soundlocity of the films. However, using the sound velocity drived from the elastic modulus measured separately usnanoindentation for each film, the results obtained show oa qualitative agreement with experiment, with errors630%. A more successful approach would not only clothe gap between model and experiment but, for technologrelevance, also include the effects of local chemical bondand deposition process methodology.

The CDO materials described above are interestingthat the network is generally ansp3 bonded one, in whichthe material density is reduced by configurational rearranment to accommodate the dopant materials. The resumolecular cage-like pores are on the atomic scale and pably do not require explicit consideration in an elementatheoretical description. However, more aggressive materthose providing a dielectric permittivity near 2, can be ppared using the explicit introduction of nanoscale porosand alternative local chemistry. The manner in which oincludes nanoscale pores into a theoretical approach for tmal transport is not yet precise. For example, the descripof the transition from ‘‘cage-like’’ to nanoporosity needsbe clarified. Perhaps more importantly, the pore sizes ofterest, which are a few nanometers so as not to limitmechanical strength of the films, are comparable to theevant thermal phonon wavelengths, which renders ctinuum approximations suspect. The effect of porosity onroom temperature thermal conductivity of xerogel, a nanorous form of a-SiO2 , was recently studied by Hu anco-workers;109 they proposed phenomenological two-phamixture models that usefully describe the dependence ofconductivity on porosity.

To explore the effect of porosity on the thermal transpin lower permittivity amorphous materials, Costescuet al.110

have investigated the thermal conductivity of hydrogesilsesquioxane~HSQ! films in the temperature range 80,T,400 K using the 3v method.13,111 HSQ is a spin-on material with dielectric permittivities near to 2. Figure 13 show

-

n


arn

ultit

n

r

ttthdna

thheeeo

iu

b

eraclyta

ius

sedsee

-

tallane

be

aveyeroft if

te-

the

theal

us

al3 tos

-eg-

hisses,edes:

ofar-tlyrved

ityity

of

ns-oreper-

ingic-y-s-ands are-

iO

le


the results of thermal conductivity measurements on a vety of HSQ samples that differ primarily by processing coditions, which is expected to modulate the porosity of resant films. Remarkably, the temperature dependence ofdata is nearly identical for all samples, i.e., the conductivof HSQ is suppressed relative to SiO2 by a constant factorthat is independent of temperature in the temperature raof the experiments, 80,T,400 K, and a significant differ-ence in porosity does not appear to modify the charactethermal transport.

Thus, the nearly identical temperature dependence ofthermal conductivities of SiO2 , FOx , and XLK suggests thathe spectrum of heat-carrying vibrational modes is nearlysame in all three materials. It is thus attractive to concluthat heat transport in these materials is intrinsically local athat porosity does not play a role except to reduce averdensity. However, this is the basic physical picture ofminimum thermal conductivity theory applied above for tdoped oxides.108 The results of this theory applied to thHSQ materials are presented in Fig. 13, where it is evidthat with increasing porosity, there is an increasingly pomatch between the data and the model. Effective medtheory ~Refs. 112 and 113! in which the porous solid istreated as a composite material of a matrix punctuatedvoids offers a path to better understand these materials.110

B. Multilayers and superlattices

Multilayers and superlattices are films that contain altnating layers of two different materials stacked upon eother. In multilayers the films may be amorphous or pocrystalline while in superlattices the films are single crysIn this section, we concentrate on the thermal conductivitythe direction perpendicular to the plane of the film beca

FIG. 13. Comparison of data for hydrogen-silsesquioxane HSQ and S2

~filled circles! to calculations of the minimum thermal conductivity~dashedlines! for each material. The mass density of the ‘‘flowable oxide’’ sampFOx-A, is r51.5 g cm23; for the ‘‘extra-low-k’’ sample, XLK-6, r50.90 g cm23.


i--t-hey

ge

of

he

eedgee

ntrm

y

-h-l.ne

this is the property accessible to the most commonly uexperimental techniques: time-domain thermoreflectance,Sec. V A, and the 3v method for thin films.111,13 ~When thewidth of the heater/thermometer line in the 3v method iscomparable to the film thickness, the 3v method is also sensitive to the thermal conductivity in the plane of the film.114

Data collected using combinations of narrow and wide melines can be used to separate the in-plane and cross-pconductivities.114!

Phonon transport in superlattices and multilayers candivided into two categories, namely:~i! ,,b; and ~ii ! ,.b, where , is the phonon mean free path andb is thesuperlattice period. Consider the first regime. There hbeen a number of studies of thermal transport in multilafilms12,14,115with heat transport perpendicular to the planethe film. The general conclusion from these studies is thathe thermal conductivity of a thin layer of disordered marial is independent of layer thicknessb, i.e., if phonons withmean-free-paths comparable tob make a negligible contri-bution to heat transport, and that the microstructure ofthin layer is approximately independent ofb, the measuredresistance of a multilayer sample is the series sum ofthermal resistance of the individual layers and the thermresistance of the interfaces.13,116Thus, a simple way of pre-dicting the thermal conductivity is to add up the bulk plinterfacial resistances of all the layers.

Figure 1 shows the data and predictions for the thermconductance of a single interface. The values range from200 MW m22 K21 at room temperature while the predictionof the diffuse mismatch model~DMM ! tend to be on theorder of 500 MW m22 K21. While this discrepancy continues to exist, in many cases the interfacial resistance is nligible compared to the bulk resistance of a single layer. Toften occurs when the films are amorphous. In such cathe thermal conductivity of multilayer films is found to bindependent of periodb. Studies of multilayers of disordereoxides have failed to reveal a significant effect of interfacthe thermal conductivity of ZrO2 :Y2O3 /SiO2 ~Ref. 117! andZrO2 /Y2O3 ~Ref. 12! multilayers are almost independentlayer thickness with layer thickness as small as 4 nm. Appently, the lattice vibrations of these oxides are sufficienalike that the thermal conductance is too large to be obseby this approach. Multilayer coatings115 do not, therefore,appear to be a promising route for lowering the conductivof thermal barriers. Significant decreases in conductivhave been observed recently in nanocrystalline ZrO2 :Y2O3

~Ref. 118! and attributed to the finite thermal conductancegrain boundaries.

Theoretical and experimental studies on thermal traport in epitaxial superlattices have been conducted for mthan two decades. In most cases, phonon transport in sulattices fall into the second regime, i.e.,,.b. Hence, bothwave and particle-like phonon effects are observed, offernot only richness in physics but also difficulties in predtions. The origins of this work can be traced back to Naraanamurtiet al.;119 they studied monochromatic phonon tranport across lattice matched GaAs/AlGaAs superlatticesshowed the existence of a phonon band gap. This icoupled behavior resulting from interference of waves

,


e tra


TABLE I. Experimental research on thermal transport in semiconductor superlattices. Unless otherwise noted, the comments refer to cross-plannsport.AM stands for ‘‘acoustic mismatch.’’

Reference Materials Effects considered Temperature behavior Period behaviorLowest thermal

conductivity

124 GaAs/AlAs not applicable not applicable in-plane: max at 200 Å in-plane: 12 W m21 K21

125 GaAs/AlAs interface roughness in-plane: decrease 190–450 K~700 Å/700 Å only! in-plane:'40133 GaAs/AlAs AM, band gaps, roughness decrease 100–400 K increase 10–400 Å 4122 Si/Ge dislocations increase to'150 K maximum at 70 Å '1.667 Si/Si0.71Ge0.29 AM not applicable 50 Å/50 Å only! 22, in-plane: 54123 Si/Ge doping, period,

dislocationsincrease 80–150 K,constant to 330 K

decreaseb540– 90 Å 1.2

slight decrease tob5140 Å127 Si/Si0.7Ge0.3 AM increase 100–300 K increase forb545– 300 Å 10

in-plane: max shifts with period134 Si0.84Ge0.16/Si0.76Ge0.24 AM increase 50–180 K,

constant to 300 Kconstant forb567, 100, 133 Å 8.4

128 InAs/AlSb growth temp, annealing maximum at 150 K not applicable 2.5135 InP/InGaAs thickness ratio increase 50–100 K not applicable 7132 Bi2Te3 /Sb2Te3 doping, tunneling,

localizationnot applicable lattice: minimum at 40 Å 0.2

12 PbTe/PbSe not applicable decrease 200–400 K 40 Å only 1.6

a

enofonu

,stap

rge

whn

areef-

er-cts.duc-rethece,

ute

ex-ed.ileostandhe

flected from multiple interfaces. For this to occur, the mefree path of phonons must span multiple interfaces (.10).In this regime, the phonon dispersion relation is modifiand zone folding occurs resulting in multiple phonon bagaps.120 Several effects can result from this modificationthe phonon dispersion. First, the group velocities of phonare reduced significantly, especially for higher energy acotic phonons. Second, becausev –k relations are modifiedthere are many more possibilities for conservation of crymomentum and energy involved in normal and umklascattering.121 Hence, the scattering rate is increased.

In the regime when the mean free path is not laenough for phonon band gaps to occur, single interfacefects can also play a significant role. For example, if the tmaterials in the superlattice have large mismatch in the pnon dispersion relations, phonons in certain frequency ra


n

dd

ss-

lp

ef-oo-ge

cannot propagate to the neighboring layer unless theremode conversions at the interface. In addition to thesefects, interfaces between two different materials with diffent lattice constants can contain dislocations and defeThese can also scatter phonons and reduce thermal contivity. Finally, depending on the type of processing, thecould be both physical roughness as well as alloying atinterface, which can also influence phonon transport. Henit is clear that there are many different effects that contribto the resistance to heat flow.

To dissect the contributions of these effects, severalperimental and theoretical studies have been performTable I lists the experimental work done in this area whTable II lists the theoretical and computational ones. Mexperimental studies have used superlattices of Si, Ge,their alloys as well as those of III–V. Table I also shows t

TABLE II. Theoretical research on thermal transport in semiconductor superlattices. AM stands for ‘‘acoustic mismatch.’’

Reference Superlattice Model Effects considered Comparison with experiment

121 general phenomenological minibands conductivity increases with period136 Si/Ge sc lattice dynamics AM, phonon confinement, minibands room temperature reduction

agrees well with Si/Ge137–139 Si/Ge;

GaAs/AlAsBoltzmann transportequation

AM, inelastic scattering, roughness see references

140 Si/Ge fcc lattice dynamics minibands, velocity trend does not agree with experimental results141 general transfer matrix phonon interference, tunneling poor agreement with experiment142 GaAs/AlAs lattice dynamics velocity reduction poor agreement with experiment143 Si/Ge valence-force potential velocity reduction, spectra mismatch periods too thin for comparison144 Si/Ge & GaAs/AlAs fcc lattice dynamics minibands, tunneling good agreement for cross-plane;

poor agreement in-plane120 general wave theory mini-bands, interface scattering trend agrees with lattice conductivity of Bi2Te3 /Sb2Te3

145 Si/Ge molecular dynamics all classical effects good agreement~two points! with experiment146 general phenomenological particle-wave crossover high TC at ultra-short periods; model

cannot explain trend147 SiGe quantum dot continuum quantum dot scattering not applicable132 Bi2Te3 /Sb2Te3 Boltzmann transport

equationAM, interface roughness,localization

trend does not agree with data

148 Lennard–Jones solids molecular dynamics all classical effects, strain not applicable149 fcc molecular dynamics all classical effects not applicable


psoioveiconinndt

ic

e

fan

eraarte

thcanyer

ofesityhe

b-her-e

gn.ho-andin

ieder-ses,

thece

al-in

int of

tic

allc

y ode

ap

of

e-

.


effects considered~interfacial roughness, phonon band gadispersion mismatch, etc.! and the observed dependencethermal conductivity on temperature and superlattice perThe data on Si and Ge superlattices suggest that forsmall period, when the layer thickness is less than the critone for the extension of misfit disclocations, the thermal cductivity of superlattices tend to decrease with decreaslayer thickness.122As the layer thickness is increased beyo10 nm, the thermal conductivity drops, presumably duedisorder created by extensive plastic deformation@see Fig.14~a!#. This is similar to the observations of BorcaTascuet al.123

Interface disorder introduces diffuse scattering but intface disorder in a superlattice is typically modest. It is limited to interface roughness~a finite density of interface steps!and substitutional alloying at the interfaces caused by sursegregation during growth. Unfortunately, this physical achemical interface roughness in an epitaxial superlatticsensitive to the material, growth method, growth tempeture, and deposition rate—and is notoriously difficult to chacterize. Furthermore, asymmetry of the interface is of

FIG. 14. Thermal conductivity of Si/Ge~top! ~Ref. 122! and Si/Si0.7Ge0.3

~bottom! ~Ref. 127! superlattices; the curves are labeled by the superlatperiod in Å. The thermal conductivity of Si/Ge superlattices~top! increaseswith period thickness in the region of 30–60 Å, but then drops dramaticfor period higher than 140 Å, presumably due to presence of misfit dislotions and defects beyond a critical thickness. The thermal conductivitSi/Si0.7Ge0.3 with thickness ratio of 2:1 showed that as the period wascreased from 300 Å~100 periods in 3mm thick film! to 45 Å ~667 periodsin 3 mm thick film!, the thermal conductivity decreased steadily andproached the effective alloy which is Si0.9Ge0.1.


,fd.ryal-g

o

r--

cedis--n

pronounced, i.e., the growth of componentA on the surfaceof componentB creates a different interface than the growof B on A. Strained-layer superlattices such as Si–Geharbor high densities of crystalline defects when the lathickness exceeds the critical thickness for the extensionmisfit dislocations; growth on relaxed buffer layers reducthe density of threading dislocations but the misfit denswill still be large when the critical thickness is exceeded. Tagreement between the data of Cahillet al.122 and BorcaTas-cuic et al.123 seem to support this idea. The interesting oservation that they both made was that the superlattice tmal conductivity was lower than that of the alloy with thsame mass ratios~see Fig. 15!. Hence, interface scatterinalone is probably insufficient to explain this observatioPerhaps this may be due to zone folding which reduces pnon group velocities and increases scattering ratesthereby reduces thermal conductivity. Thermal transportGaAs and AlAs superlattices have also been studwidely.66,124–126These studies have shown that as the intface density increases, the thermal conductivity decreaclearly showing that interface resistance is important.

More recently, Huxtableet al.127 have shown that forsuperlattices of Si and Si0.7Ge0.3, where the critical thicknessis much larger than that for Si and Ge superlattices,thermal conductivity scaled almost linearly with interfadensity supporting the data of Cahillet al.122 andBorcaTascuic123 @see Fig. 14~b!#. As the interface density in-creased, the thermal conductivity approached that of theloy, but was never lower than that of the alloy, which iscontrast with previous observations. Clearly, the mismatchacoustic impedance between Si and Ge is more than tha

e

ya-f-

-

FIG. 15. Selected data for the through-thickness thermal conductivitysuperlattices with bilayer periods of'5 nm; data for GaAs–AlAs@opencircles, 5.67 nm period~Ref. 126!# were measured by picosecond thermorflectance; data for Si–Ge@open triangles, 5 nm period~Ref. 122!; filledcircles, 4.4 nm period~Ref. 123!# and InAs–AlSb@open diamonds, 6.5 nmperiod~Ref. 128! were measured using the 3v method. Data for Si0.85Ge0.15

~Ref. 122! and Ga0.4Al0.6As ~Ref. 129! alloys are included for comparison


c-is

ca

oera

n-haalg

ictt

ongthltdy

earlae

doajth

tioe

neeahechnbdty,phFolloera

nete

tticir

ightinha

bh

llelortr-

arlynon

yer.totor

eof

toic

ters

en-s-

maly

s aonated

uc-tod on

.ort

love-ousin

fac-er-tal

nthethees.c-

on-z-

ndom

u-


Si and Si0.7Ge0.3. Hence, it is possible that thermal condutivities below alloy values can be found only when therelarge mismatch in acoustic impedance that creates signifiband gaps in the phonon dispersion relation.

There have been two reports of thermal conductivityBi2Te3 and Sb2Te3 superlattices. These are of interest in thmoelectric applications where the reduction of thermconductivity has been a major challenge.130

Venkatasubramanian131 showed as the period thickness icreased, the room temperature thermal conductivity reaca minimum for a period thickness 5 nm, and this value wlower than the thermal conductivity of the correspondingloy. It was suggested that this was due to phonon band-effects. However, Touzelbaevet al.132 did not find any sig-nificant change in thermal conductivity when the superlattperiod was varied. Hence, it remains inconclusive aswhether phonon band-gap effects are indeed present insystem.

Table II lists the theoretical and computational workthermal transport in superlattices. Three approaches areerally adopted. The first is based on wave propagation,second on particle transport that can be modeled by Bomann transport theory, and the third relies on molecularnamics.

From the experimental and theoretical studies, it is clthat the mechanism of phonon transport in epitaxial supetices is not well understood and it is difficult to predict thvalue of thermal conductivity. However, a few trendsseem to appear. First, if interface resistance is to play a mrole, the acoustic mismatch between the two films ofsuperlattice must be sufficiently large,.1.1. For incoherentphonons, acoustic mismatch gives rise to interface reflecwhile for coherent phonons, phonon band gaps are formthat reduce the group velocities. Second, interface roughand alloying can also scatter phonons and thereby incrinterfacial resistance. Third, in some cases, the phonon tmal conductivity can be reduced below the alloy limit, whimay occur due to band-gap formation. Although calculatiopredict this, experiments have not provided unmistakaproof. The image that seems to emerge from all these stuis the following. To reach the minimum thermal conductivione perhaps needs to take different approaches to blocknon transport in different parts of the phonon spectrum.example, high-frequency phonons can be blocked by ascattering since the wavelengths are generally on the orda few atomic spacings. Low frequency phonons are generunaffected by alloy scattering since the scattering is geally in the Rayleigh regime. However, they may dominaenergy transport in an alloy. For such phonons, superlaformation would lead to miniband formation since themean free paths are generally much larger than for hfrequency phonons. Hence, what would indeed be interesis to study thermal transport in superlattices of alloys thave sufficient acoustic impedance mismatch.

C. Superlattices: Theory and computation

Thermal transport in semiconductor superlattices caneither parallel or perpendicular to the plane of the layers. T


nt

f-l

eds-ap

eohis

en-e

z--

rt-

ore

ndssser-

sleies

o-ryof

llyr-

e

-gt

ee

initial measurements and theories were for transport parato the layers. The simple viewpoint is that heat transpshould be quite efficient parallel to the layers. If the intefaces are atomically smooth, then phonons would speculreflect from these interfaces. Each layer becomes a phowaveguide which efficiently channels heat along each laThe initial measurements of thermal conductivity parallelthe layers found values smaller than bulk values by a facof four. Yao124 found in GaAs/AlAs superlattices that ththermal conductivity was similar to that of a random alloythe two materials. This was explained in Ref. 150 as duethe interface boundaries not being perfectly planar. Atomscale defects at the interface are efficient scattering cenfor the phonons.

There is much more interest in thermal transport perpdicular to the plane of the layers, which is called the ‘‘crosplane’’ direction. In some cases experiments show a therconductivity significantly lower than that of the random alloof the two materials in the superlattice.122,131,133The simpleexplanation for this reduction is that the superlattice acts aFabrey–Perot interferometer, and allows only a few phonfrequencies to pass through. This feature was demonstrby phonon transmission spectroscopy.119

There have been many calculations of thermal condtivity in the cross plane direction. They can be divided inthree groups. The largest number of calculations are basethe standard formula for the thermal conductivity

k5\(lE d3q

~2p!3 vzl~q!2vl~q!tl~q!S dnB@vl~q!#

dT D ,

~8!

where vzl(q) is the velocity of the phonon,tl(q) is thelifetime, andnB@vl(q)# is the Boson occupation functionThe above formula is derived from the Boltzmann transpequation~BTE!. Calculation by Kato et al,151 Hyldgaard andMahan,136 Tamura et al.,140 Simkin and Mahan,120 Bieset al.,142 and Kiselevet al.,143 all just calculate the actuaphonon modes of the superlattice and evaluate the abformula. Usually the lifetimetl(q), or else the mean-freepath ,lq is selected to be the same as in the homogenematerial. These calculations show the thermal conductivitythe superlattice is reduced significantly, sometimes by ator of ten, compared to that of the constituents of the suplattice. This large reduction agrees with the experimenfindings.

Simkin and Mahan120 used a complex wave vector icomputing the superlattice modes. The imaginary part ofwave vector was a phenomenological way of includingmean-free-path into the calculation of the phonon modThis calculation showed a minimum in the thermal condutivity as a function of superlattice period.

There have been several calculations of the thermal cductivity of the superlattice which have solved the Boltmann transport equation. That of Ren and Dow121 did notpredict a large reduction of the thermal conductivity, adisagrees with experiments. Several calculations frChen’s group138,152agree much better with experiment.

Recently, Daly and co-workers149,153 used large-scale(1.33106 atoms) molecular dynamics simulations to calc


ona

ana

gonb

urf

oyaruheeivl-uanwinersilaune

rtiFosalthmria

ret

eno

anaf

adinls

tie, rccsu

thein-

lleralare

ibed

ondaindif-

teri-

-tioniso-al

ce

alits

re-ed

od-

cal, orofs a

thens inity

useory.


late the thermal conductivity of superlattices.There have been very few calculations of thermal c

ductivity in the cross plane direction which included hetransfer by electrons as well as phonons. BartkowiakMahan’s modeling146 was semiclassical and assumed thboth the electronsTe(x) and phononsTp(x) had a local tem-perature. The electrons and phonons can exchange enerthe bulk, and at the interfaces. They found a wide varietytemperature distributions depending upon the various bouary resistances for electrons and phonons. None of thishavior has been verified or refuted by experimental measments. All of their calculations depend on the concept olocal temperature.

V. NEEDS AND LIMITS OF METROLOGY

The goal of successful commercial implementationnanoscale materials places a burden on the metrologthese materials and devices that extend beyond the chterization of the fundamental materials properties. This bden involves not only the small physical dimension of tmaterial but also the general accuracy of the measuremTo illustrate this, perhaps the most familiar examples derfrom the Si semiconductor industry. Here, thin film metroogy is becoming increasingly challenging as Si semicondtor device technology scales toward smaller dimensionsthe corresponding manufacturing process complexity groTraditional thin film properties such as thickness and intrsic stress are being augmented by an increasing numbother physical parameters, which together must be conered in the development and manufacturing monitoring pSuch parameters include density, porosity, thermal condtivity, grain texture, surface chemical residue, and bulk ainterface cohesion. The introduction of more complex dvices using new materials and process methods~e.g., Si al-loys, polymers, and 3D structures! also contribute to themeasurement challenge. Effective control of these propeoften becomes difficult as the device scaling continues.example, the control of the physical thickness and comption profile of the MOS gate dielectric is generally of criticimportance to the resultant device parameters. But withfilm currently having a required thickness of less than 1 nthe challenge is to control thickness and composition vation on the atomic scale.

Another factor to consider is that the ability to measumany properties is becoming less and less feasible atnanoscale limit. A well known example is the measuremof dopant distributions under a gate as the gate widths mbelow the 50 nm scale and approach the ‘‘sparse doplimit.154As a result, an increasing number of traditional mesurement methods are becoming insensitive or unsuitableuse due to the limitations of the physics employed. Todress these difficulties, not only is new instrumentation besought, but also industry is relying increasingly on materiadevice, and thermal-mechanical modeling to infer properthat cannot be directly measured. This approach, in turnquires that the fundamental properties be precisely and arately measured on test systems, designed to enablemeasurements.


-tdt

y infd-e-e-a

fofac-r-

nt.e

c-d

s.-of

d-n.c-d-

esori-

is,-

het

vet’’-or-g,se-u-ch

The consequence of the above considerations forproblem of heat transport in nanostructures has been thetroduction of a wide variety of novel instrumentation wesuited for measurement of thin films and structures. Sevpromising approaches have been introduced recently orcurrently under development. Three examples are descrbelow.

A. Time-domain thermoreflectance

The first thermal transport experiments using picosecthermoreflectance, alternatively referred to as time-domthermoreflectance~TDTR!, were reported in Refs. 155 an156. TDTR provides a direct method for measuring heat dfusion on nanometer length scales. Most engineering maals have thermal diffusivitiesD in the range 0.005,D,1 cm2 s21 and at t5200 ps, heat diffusion lengthsl5ADt are in the range 10, l ,140 nm. Therefore, picosecond time resolution offers nanometer-scale depth resoluand, more specifically, picosecond thermoreflectance canlate the effects of interface conductance from the thermconductivity of a thin layer. By contrast, thermoreflectanmeasurements using longer pulse lengthQ-switched lasers157

and the 3v method13 cannot distinguish between the thermconductivity of a film and the thermal conductance ofinterfaces.

An example of TDTR experimental apparatus is psented in Fig. 16. Picosecond time resolution is obtainusing mode-locked lasers that produce a series of,1 pspulses at a repetition rate of'76 MHz. Extremely high timeresolution is achieved by splitting the laser output into twbeam paths, a ‘‘pump’’ beam and a ‘‘probe’’ beam, and ajusting the relative optical path lengths with a mechanidelay stage. If the sample under investigation is metalliccan be coated with a thin metal film, then a small fractionthe energy from each pulse in the pump beam producesudden jump in temperature of;3 K near the surface of thesample. The decay of the near-surface temperature isinterrogated by the reflected energy of the series of pulsethe probe beam. Determination of the thermal conductiv

FIG. 16. Schematic of the TDTR and picosecond acoustics apparatus inat the Laser Facility of the Frederick Seitz Materials Research Laborat


thopeo

incth

cepeeisese-vehy-

tivia

a

ole

th

srerla

cuthtid

anowoe

ile

dethfR

asb

slmpr

tu-alap-mi-ica-

ity

tem-e, aout.le,ple

d forp-

peanti-anti-. To-rmalolidthepera-is-,


and interfacial thermal resistance is made by comparingexperimental cooling curve to the theoretical model andtimizing the free parameters. For all but the shortest timthe temperature is homogeneous through the optical abstion depth of the thin metal film and DR1(t)5(dR/dT)DT1(t). The surface temperatureDT1(t) can beaccurately calculated using one-dimensional heat flow sthe thermal diffusion length is usually much smaller thanradius of the focused pump beam.133

In a modulated pump–probe experiment, the differenin reflected probe intensity caused by the pump pulse apat the modulation frequency of the pump beam and aretracted with lock-in detection but interpretation of thlock-in signal is not straightforward as the time delay bcomes a significant fraction of the separation between pulCapinski and co-workers133 discuss calculations of the inphase and out-of-phase signals of the lock-in amplifier githe time-domain thermal response of the sample; Caet al.1 provide a calculation starting from the frequencdomain response.

Despite the promise of these methods, few quantitaand systematic studies of thermal properties of materhave used picosecond optical techniques.10,11,133,156,158,159

Experiments by Maris and co-workers on interface thermconductance10 and superlattice conductivity133 are the mostcomplete. Data for the thermal decay at short timest,50 ps are difficult to interpret quantitatively because~i!hot-electrons can deposit energy beyond the optical abstion depth,~ii ! the temperature dependence of the compindex of refractiondn/dT is typically unknown,~iii ! the dif-fusion equation fails on length scales comparable tomean-free-paths of the dominant energy carriers, and~iv! theassumption of equilibrium between phonons and electronnot always valid. At long timest.500 ps, measurements aoften plagued by shifts in the focal plane and beam ovecreated by large displacements of the delay stage.~Capinskiand Maris104 describe an optical design that improves acracy at long delay times; the out-of-phase signal oflock-in amplifier can also be used to correct systemaerrors.1! It is likely, however, that improved performance andropping costs of solid-state mode-locked lasers will expapplications of picosecond thermoreflectance. Reliable, lnoise Ti:sapphire lasers that produce 100 fs pulses are cmercially available and can be operated with only a modknowledge of the technology of ultrafast lasers.

Over the longer term, improved laser performance walso make more sophisticated experiments possible. Forample, the pump and probe light pulses can be applielocalized regions of the sample through the use of near fioptics. With this technique, it should be possible to studylateral flow of heat in a thin film, instead of just the flow oheat through the film. In addition, the accuracy of TDTmeasurements will be enormously enhanced if compact lsources that can be tuned over a wide wavelength rangecome available. The probe wavelength could then be choto match a peak in the thermoreflectance of the metal fiThis would increase signal to noise in the measurementcess, reduce the time needed for a measurement, andprove the accuracy.


e-

s,rp-

ee

sarx-

-s:

nill

els

l

rp-x

e

is

p

-ec

d-

m-st

lx-tolde

ere-

en.

o-im-

B. Microfabricated probes

Access to nanofabrication facilities offers new oppornities for probing directly the local temperature and thermtransport mechanisms in nanostructures. Two significantproaches will be commented on below: scanning thermalcroscopy and the new area of microinstrumentation fabrtion.

A scanning thermal microscope~SThM! operates bybringing a sharp temperature-sensing tip in close proximto a sample solid surface~see Fig. 17!. Localized heat trans-fer between the tip and sample surface changes the tipperature. By scanning the tip across the sample surfacspatial map of the tip-sample heat transfer is mappedWhen the tip comes in local equilibrium with the sampone obtains the spatial temperature distribution of the samsurface, whereas if the temperature change is determinea known heat flux, one could obtain the local thermal pro

FIG. 17. ~Color! Schematic diagram of a scanning thermal microsco~SThM!. It consists of a sharp temperature-sensing tip mounted on a clever probe. The sample is scanned in the lateral directions while the clever deflections are monitored using a laser beam-deflection techniquepographical and thermal images can be thermally obtained. The thetransport at the tip-sample contacts consists of air, liquid, and solid–sconduction pathways. A simple thermal resistance network model ofsample and probe combination, shows that when the sample is at temture Ts , the tip temperatureTt depends on the values of the thermal restances of the tip-sample contact,Rts , the tip,Rt , and the cantilever probeRc .


otf


FIG. 18. ~Color! Topographical~top! and thermal images of a multiwall carbon nanotube~MWCN!, 10 nm in diameter obtained by SThM. The bottom plshows the temperature distribution along the length of the wire while the plot on the side shows the cross-sectional temperature profile, indicatingull-widthhalf maximum of the temperature peak to be 50 nm.

toist

inng

ndhepyns

t pabyonpata

g

ilns

sedalo-

er--

uresgher-on as-d

on-eleera-s in

ures-eve-

aterion,er-

erties. The spatial resolution depends mainly on three facnamely, tip sharpness, tip-sample heat transfer mechanand the thermal design of the probe. In the recent past,spatial resolution has been improved to 30–50 nm, allowone to study thermal transport phenomena at these lescales.160

The idea of SThM was first proposed by Williams aWickramasinghe161 when they used a thermocouple at tend of a metallic wire to measure temperature. Tip-samheat transfer was used to image the surface topographelectrically insulating materials. Since then, significaprogress has been made in improving thermocouple-bameasurements while other techniques based on contactential, electrical resistance, and thermal expansion havebeen developed, which are reviewed in detailMajumdar.162 Most of the development has focusedcantilever-based probes such that atomic force microsco~AFMs!163 could be used as a platform for SThM. Whentemperature sensor is mounted on the very apex of thesuch probes can be used to image both the topographythe temperature distribution of devices such as sintransistors26,77and vertical cavity lasers.164 Such studies havehelped in identifying defects and failure mechanisms whproviding insight about electron, photon and phonon tra


rs,m,hegth

leoftedo-

lso

es

ip,nd

le

e-

port in these devices. In addition, SThM has also been uto measure thermal properties of materials and perform crimetry at nanometer scales.165

There are three important elements determining the pformance of SThM, namely:~i! thermal and mechanical design of the probe;~ii ! fabrication of probe;~iii ! understand-ing of tip-sample heat transfer. Proper thermal design ensnot only temperature measurement accuracy but also hispatial resolution.166,167Fabrication of probes by MEMS processes allows one to batch fabricate hundreds of probessingle wafer using optical lithography, enabling multiple uers to utilize such probes.160,166Recent studies have revealethe role of gas conduction, solid–solid conduction, and cduction through a liquid film bridging the tip and thsample.164,167A fundamental understanding of the tip-sampheat transfer allows accurate estimation of sample tempture distributions from the measured data. Recent progresthese areas has allowed thermal imaging of nanostructsuch as carbon nanotubes160 with 30–50 nm spatial resolution ~see Fig. 18!. Such a high spatial resolution allows onto study phonon physics at the length scale of phonon walengths and mean free paths, which could lead to greunderstanding of nanoscale thermal transport. In additthe spatial resolution is sufficiently high to study electroth


to

nwb-fecep-n

oiwr

esdarplrd

owipliveti

teth

erreolsenere

uidins

le.o-a

uclly-pto

oretum

incaistSanponm

uc-ros-p isig-

icro-MSanrop-of-

bees

anthe

ns.rentter-ionnonthat

ble

icu-

eping


mal and photothermal effects in modern electronic and opelectronic devices.

Despite these recent advances, there are several uswered questions that need to be resolved. It is well knothat the liquid film bridging the tip and the sample contriutes significantly to tip-sample heat transfer. Can the efof liquid conduction be controlled by tailoring the surfachemistry? What is the role of near-field radiation in tisample heat transfer, especially in a vacuum environmeHow do phonons and electrons propagate through a pcontact that is on the order of their wavelength? Canconduct SThM experiments at low temperature and obsequantum transport effects? Because tip-sample thermal rtance depends on surface topography, topography-relatetifacts in thermal images still remains a problem. Theretwo potential solutions: a thorough knowledge of tip-samheat transfer in order to model the effect of topography; onull-point measurement such that tip-sample temperatureference can be minimized thereby eliminating any heat flThe latter requires more sophisticated probes with multsensors and feedback control. Future progress will be drtowards multiple sensors and actuators on integrated canver probes such that one could simultaneously measureperature and heat flux as well as electrical, optical, and onanoscale properties while imaging a nanostructure.

Finally, because new nanoscale materials and enconversion devices are being developed, there will alsomain a need to thermally probe them at nanoscales. Forample, thermoelectric properties of nanostructured semicductors are markedly different than those of bulk materiaBy simultaneously measuring thermal, electrical, and thmoelectric properties using a multifunctional SThM, ocould characterize low-dimensional nanostructures. An aof research that is largely unexplored is SThM under liqenvironment, in particular, of biological molecules. Proteand DNA undergo structural changes and phase transitiondifferent temperatures. Localized calorimetry of molecucould shed light on the binding behavior of biomolecules

One of the major limitations of scanning probe micrscopes is their inability to probe below a solid surface. Blistic electron emission microscopy~BEEM! allows one toprobe about 10-50 nm below a conducting or a semiconding surface by detecting electrons that travel ballisticathrough the material.168 However, an insulating surface cannot be probed by this technique. Because thermal transoccurs in all materials, it may be possible to use SThMprobe materials below the surface using either ballisticdiffusive phonon transport. While diffusive phonon transphas been explored,165 ballistic phonon transport has not ybeen studied using SThM. A combination of picosecondtrasonics and SThM might perhaps be able to map outterials and their properties in three dimensions.

The SThM probe falls under a general class of microstruments that allows one to thermally probe at a length sthat is an order of magnitude smaller than the characterdimension of the instrument. By using traditional MEMfabrication techniques, which allow one to make devicesstructures down to about 300 nm, one could study transphenomena at length scales on the order of 10–50


-

an-n

ct

t?nteveis-ar-eeaif-.en

le-m-er

gye-x-n-.

r-

a

sat

s

l-

t-

ortort

l-a-

-leic

drt.

Hence, MEMS provides a platform to integrate nanostrtures and thereby forms a bridge between nano- and miccales. In the case of the SThM probe, the nanoscale tiintegrated onto the rest of the cantilever and the probe. Fure 8 shows another example where two suspended mheaters are fabricated on a chip using standard MEtechniques.94 A carbon nanotube or any 1D nanostructure cbe placed across the two heater islands and its thermal perties can be measured using this device. An advantagefered by MEMS is that hundreds of microinstruments canbuilt on a single wafer, providing low cost measuring devicand high-throughput experimentation.

C. Coherent optical methods

Coherent optical methods, such as stimulated Ramscattering, have long been applied to measurements oflifetime of optical phonons.169 As illustrated in Fig. 19, laserexcitation can lead to the generation of intense vibratioshowing a high degree of temporal and spatial coherence170

Transport measurements using optically generated cohephonons, particularly acoustic modes, may provide an alnative to conventional thermal methods for the determinatof the mean free path. Coherent, monochromatic phobeams can be generated using a variety of techniquescombine optical cw or pulsed lasers and a suita

FIG. 19. Normalized differential reflectivity for antimony at 300 K~see Ref.170!. Ec is the central energy of the pulses which are polarized perpendlar to the trigonal axis. The data show coherentA1g phonon oscillations offrequencyV0;4.5 THz and a slowly varying electronic background. Thfine structure near zero delay is due to interference between overlappump and probe beams.


aly

ngm

reÅh,

usphseluid0

gr

aueenousndde

c—mnita

asan

oydicu

plcp-

o

loba

iarg

eterantE-

t ofi-u-aveS.o-elpAird

De-

R-rchRItante-ndent9-

. A

.

ys.

s.

e,

tt.

ett.

S.


transducer.170–173 Laser-induced cw thermomodulation ofthin metallic film gives a tunable source of longitudinalpolarized phonons in the 1–5 GHz range.171 Higher frequen-cies, up to;0.5 THz, can be attained using ultrafast~subpi-cosecond! laser excitation of a layered structure.172 In semi-conductor superlattices, the mechanism for light-soucoupling is the acousto-optic effect, and phase matchinprovided by the wave vectors involved in the Fourier decoposition of the structure.170

As discussed earlier, it is possible to generate cohe~but not monochromatic! sound pulses as short as 100using picosecond ultrasonic methods. This technique

been recently applied to studies of the phonon dispersion173

and measurements of the phonon attenuation could beto determine the mean free path. We note that coherentnon sources can potentially be used for microscopy purpoHere, there is considerable room for improving the resotion. While the wavelengths of propagating modes in solare small enough to resolve objects separated by, say, 10the length scale for state-of-the-art ultrasound and tomophy methods is 2–3 orders of magnitude larger.174

VI. OUTLOOK

The many unresolved fundamental issues; continuingvances in theory, computation, and experimental techniqand important applications in modern device materialssure that nanoscale thermal transport will remain a vigorfield of investigation for many years to come. We discuspecific examples of outstanding questions and recommetions for future directions throughout this review but provia brief summary of some selected examples here.

The limited range of the data for the thermal condutance of solid–solid interfaces is not well understoodtheory predicts a much stronger dependence on acousticmatch than what is observed. Transport and coherent phoexperiments on individual and superlattice interfaces wsystematically controlled and well-characterized structureneeded. A simple, analytical model that captures the bfeatures of the frequency dependence in the phonon trmission coefficient would be of great use. Large-scale mlecular dynamics should be used to evaluate the sensitivitthe thermal conductance to the structural and chemicalorder of the interface. A long term goal is to include eletronic and quantum mechanical effects in atomic-level simlations of thermal transport.

Another important long term challenge is the develoment of techniques that incorporate scattering rates calated by atomic-level simulations into higher-level aproaches, e.g., Monte Carlo solutions of the classicalquantum Boltzmann transport equation. Improvementsscanning thermal microscopy and other nanoscale metromethods are needed to make quantitative comparisonstween experimental observations of nanoscale transportthe predictions of theory.

ACKNOWLEDGMENTS

This report was sponsored by the Council of MaterScience and Engineering of the U.S. Department of Ene


dis-

nt

as

edo-s.-sÅ,a-

d-s;-s

sa-

-

is-onhreics--

ofs---

-u-

ringye-nd

ly.

The authors thank Dr. Iran Thomas and Professor PFlynn for their support. D.G.C. was supported by NSF GrNo. CTS 99-78822 and the U.S. Department of Energy DFG02-01ER45938. A.M. greatly appreciates the supporthe NSF, the DOE~Engineering Division, Basic Energy Scences!, and DARPA and acknowledges the many contribtions of former and current students and post-docs that hcontributed in his group, in particular, A. Abramson andHuxtable for gathering some of the material for this mongraph. He also thanks P. McEuen and his group for their hwith nanotube research. H.J.M. was supported by theForce Office of Scientific Research MURI Grant entitlePhonon Enhancement of Electronic and Optoelectronicvices ~Grant No. F4962-00-1-0331!. R.M. was supported bythe National Science Foundation under Grant No. DM9876862 and by the Air Force Office of Scientific Reseaunder Contract No. F49620-00-1-0328 through the MUProgram. S.R.P. is pleased to acknowledge the imporcontributions of P. K. Schelling to this work. He also gratfully acknowledges useful discussions with P. Keblinski aJ. A. Eastman. S.R.P. was supported by the U.S. Departmof Energy, Office of Science under Contact No. W-31-10ENG-38.

1D. G. Cahill, K. E. Goodson, and A. Majumdar, J. Heat Transfer124, 223~2002!.

2I. M. Khalatnikov, Sov. Phys. JETP22, 687 ~1952!.3E. T. Swartz and R. O. Pohl, Rev. Mod. Phys.61, 605 ~1989!.4P. L. Kapitza, J. Phys.~Moscow! 4, 181 ~1941!.5L. J. Challis, K. Dransfeld, and J. Wilks, Proc. R. Soc. London, Ser260, 31 ~1961!.

6J. Weber, W. Sandmann, W. Dietsche, and H. Kinder, Phys. Rev. Lett40,1469 ~1978!.

7J. R. Olson and R. O. Pohl, J. Low Temp. Phys.94, 539 ~1994!.8H. Kinder and K. Weiss, J. Phys.: Condens. Matter5, 2063~1993!.9T. Nakayama, inProgress in Low Temperature Physics, edited by D. F.Brewer ~North-Holland, Amsterdam, 1989!, p. 115.

10R. J. Stoner and H. J. Maris, Phys. Rev. B48, 16373~1993!.11A. N. Smith, J. L. Hostetler, and P. M. Norris, Microscale Thermoph

Eng.4, 51 ~2000!.12D. G. Cahill, A. Bullen, and S.-M. Lee, High Temp.-High Press.32, 135

~2000!.13S.-M. Lee and D. G. Cahill, J. Appl. Phys.81, 2590~1997!.14E.-K. Kim, S.-I. Kwun, S.-M. Lee, H. Seo, and J.-G. Yoon, Appl. Phy

Lett. 76, 3864~2000!.15K. H. Wichert, C. Reimann, G. Blahusch, and H. Kinder, Ann. Phys.4,

175 ~1995!.16J. P. Wolfe,Imaging Phonons~Cambridge University Press, Cambridg

1998!.17M. Msall, W. Dietsche, K. J. Friedland, and Q. Y. Tong, Phys. Rev. Le

85, 598 ~2000!.18H.-Y. Hao, D. K. Sadana, and H. J. Maris, Electrochem. Solid-State L

1, 54 ~1998!.19H. T. Grahn, H. J. Maris, and J. Tauc, IEEE J. Quantum Electron.25,

2562 ~1989!.20C. J. K. Richardson, M. J. Ehrlich, and J. W. Wagner, J. Appl. Phys.85,

861 ~1999!.21A. Bartels, T. Dekorsy, H. Kurz, and K. Ko¨hler, Appl. Phys. Lett.72,

2844 ~1998!.22C. K. Sun, J. C. Liang, C. J. Stanton, A. Abare, L. Coldren, and

DenBaars, Appl. Phys. Lett.75, 1249~1999!.23U. Ozgur, C.-W. Lee, and H. O. Everitt, Phys. Rev. Lett.86, 5604~2001!.24H. S. Carslaw and J. C. Jaeger, inConduction of Heat in Solids~Oxford

University Press, New York, 1959!, pp. 109–112.25J. M. Ziman,Electrons and Phonons~Clarendon, Oxford, 1960!.26A. Majumdar, J. Heat Transfer115, 7 ~1993!.27P. B. Allen and J. L. Feldman, Phys. Rev. B48, 12581~1993!.


c

m.

s

sis

a

Kro

U.

Ds-

d-

J

S

der-

M

. B

son,

Sci-

v. B

Lett.

re

ct.

ate

.

E.

ly,ct

C.

G.

-

m.


28D. C. Rapaport,The Art of Molecular Dynamics Simulation~CambridgeUniversity Press, Cambridge, 1995!.

29P. K. Schelling, S. R. Phillpot, and P. Keblinski, Phys. Rev. B65, 144306~2002!.

30W. E. Pickett, J. L. Feldman, and J. Deppe, Modell. Simul. Mater. SEng.4, 409 ~1996!.

31A. Maiti, G. D. Mahan, and S. T. Pantelides, Solid State Commun.102,517 ~1997!.

32P. K. Schelling, S. R. Phillpot, and P. Keblinski, Phys. Rev. B65, 144306~2002!.

33P. Keblinski, S. R. Phillpot, D. Wolf, and H. Gleiter, J. Acoust. Soc. A80, 717 ~1997!.

34C.-W. Nan, R. Birringer, D. R. Clarke, and H. Gleiter, J. Appl. Phys.81,6692 ~1997!.

35E. H. Kennard,Kinetic Theory of Gases~McGraw-Hill, New York,1938!.

36W. G. Vincenti and C. H. Kruger,Introduction to Physical Gas Dynamic~Krieger, Malabar, FL, 1975!.

37R. Siegel and J. R. Howell,Thermal Radiation Heat Transfer~Hemi-sphere, New York, 1972!.

38M. Q. Brewster,Thermal Radiative Transfer and Properties~Wiley, NewYork, 1992!.

39S. Rosseland,Theoretical Astrophysics: Atomic Theory and the Analyof Stellar Atmospheres and Envelopes~Clarendon, Oxford, 1936!.

40D. A. Young and H. J. Maris, Phys. Rev. B40, 3685~1989!.41G. Fagas, A. G. Kozorezov, C. J. Lambert, and J. K. Wigmore, Physic

263–264, 739 ~1999!.42P. K. Schelling, S. R. Phillpot, and P. Keblinski, Appl. Phys. Lett.80,

2484 ~2002!.43H. B. G. Casimir, Physica5, 595 ~1938!.44C. Oligschleger and J. C. Scho¨n, Phys. Rev. B59, 4125~1999!.45B. C. Larson, J. Z. Tischler, and D. M. Mills, J. Mater. Res.1, 144~1986!.46M. L. Huberman and A. W. Overhauser, Phys. Rev. B50, 2865~1994!.47A. V. Sergeev, Phys. Rev. B58, R10199~1998!.48G. D. Mahan,Many-Particle Physics~Kluwer-Plenum, Dordrecht, 2000!.49W. R. Frensley, Rev. Mod. Phys.62, 745 ~1990!.50J. R. Hellums and W. R. Frensley, Phys. Rev. B49, 2904~1994!.51Special issue, IEEE Trans. Electron Devices45, ~1998!.52J. A. Davis, R. Venkatesan, A. Kaloyeros, M. Beylansky, S. J. Souri,

Banerjee, K. C. Saraswat, A. Rahman, R. Reif, and J. D. Meindl, PIEEE 89, 305 ~2001!.

53W. P. King, T. W. Kenny, K. E. Goodson, G. Cross, M. Despont,Durig, H. Rothuizen, G. K. Binnig, and P. Vettiger, Appl. Phys. Lett.78,1300 ~2001!.

54P. D. Vu, Xiao Liu, and R. O. Pohl, Phys. Rev. B63, 125421~2001!.55M. Asheghi, K. Kurabayashi, K. E. Goodson, R. Kasnavi, and J.

Plummer, inProceedings of the 33rd ASME/AIChE National Heat Tranfer Conference~Asme, New York, 1999!.

56M. Von Arx, O. Paul, and H. Baltes, J. Microelectromech. Syst.9, 136~2000!.

57S. Uma, A. D. McConnell, M. Asheghi, K. Kurabayashi, and K. E. Gooson, Int. J. Thermophys.22, 605 ~2001!.

58A. D. McConnell, U. Srinivasan, M. Asheghi, and K. E. Goodson,Microelectromech. Syst.10, 360 ~2001!.

59C. J. Glassbrenner and G. A. Slack, Phys. Rev. A134, 1058~1964!.60M. Asheghi, M. N. Touzelbaev, K. E. Goodson, Y. K. Leung, and S.

Wong, J. Heat Transfer120, 30 ~1998!.61M. G. Holland, Phys. Rev.132, 2461~1963!.62J. M. Ziman,Electrons and Phonons~Oxford University Press, Oxford,

1960!.63K. E. Goodson, J. Heat Transfer118, 279 ~1996!.64Y. S. Ju and K. E. Goodson, Appl. Phys. Lett.74, 3005~1999!.65R. A. Guyer and J. A. Krumhansl, Phys. Rev.148, 766 ~1966!.66G. Chen and C. L. Tien, J. Thermophys. Heat Transfer7, 311 ~1993!.67G. Chen, S. Q. Zhou, D.-Y. Yao, C. J. Kim, X. Y. Zheng, Z. L. Liu, an

K. L. Wang, inProceedings from 17th International Conference on Thmoelectrics~IEEE, Piscataway, NJ, 1998!, pp. 202–205.

68C. Moglestue, Comput. Methods Appl. Mech. Eng.30, 173 ~1982!.69C. Jacoboni and L. Reggiani, Rev. Mod. Phys.55, 642 ~1983!.70P. Lugli, P. Bordone, L. Reggiani, M. Reiger, P. Kocevar, and S.

Goodnick, Phys. Rev. B39, 7852~1989!.71M. V. Fischetti and S. E. Laux, Phys. Rev. B38, 9721~1988!.72M. V. Fischetti and S. E. Laux, Phys. Rev. B48, 2244~1993!.


i.

B

.c.

.

.

.

.

73T. Klistner, J. E. VanCleve, H. E. Fischer, and R. O. Pohl, Phys. Rev38, 7576~1988!.

74R. B. Peterson, J. Heat Transfer116, 815 ~1994!.75S. Mazumder and A. Majumdar, J. Heat Transfer123, 749 ~2001!.76M. G. Holland, Phys. Rev.134, A471 ~1964!.77J. Lai and A. Majumdar, J. Appl. Phys.79, 7353~1996!.78G. D. Mahan, J. Appl. Phys.58, 2242~1985!.79G. Chen, J. Heat Transfer118, 539 ~1996!.80P. G. Sverdrup, Y. S. Ju, and K. E. Goodson, J. Heat Transfer123, 130

~2001!.81P. G. Sverdrup, S. Sinha, M. Asheghi, U. Srinivasan, and K. E. Good

Appl. Phys. Lett.78, 3331~2001!.82K. Seeger,Semiconductor Physics, Springer Series in Solid State

ences~Springer, Berlin, 1985!, Vol. 40, p. 198.83D. D. Kim and P. Y. Yu, Phys. Rev. Lett.64, 946 ~1990!.84B. Jusserand and M. Cardona, inLight Scattering in Solids V, Topics in

Applied Physics, edited by M. Cardona and G. Gu¨ntherodt ~Springer,Heidelberg, 1989!, Vol. 66, p. 49.

85O. Madelung,Introduction to Solid State Theory~Springer, Berlin, 1978!,p. 314.

86K. Mizoguchi, M. Hase, S. Nakashima, and M. Nakayama, Phys. Re60, 8262~1999!.

87P. Bhattacharya,Semiconductor Optoelectronic Devices~Prentice-Hall,Upper Saddle River, NJ, 1997!.

88J. Urayama, T. B. Norris, J. Singh, and P. Bhattacharya, Phys. Rev.86, 4930~2001!.

89K. Schwab, E. A. Henricskon, J. M. Worlock, and M. L. Roukes, Natu~London! 404, 974 ~2000!.

90S. Berber, Y.-K. Kwon, and D. Tomanek, Phys. Rev. Lett.84, 4613~2000!.

91M. A. Osman and D. Srivastava, Nanotechnology12, 21 ~2001!.92L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B47, 16631~1993!.93J. Hone, M. Whitney, C. Piskoti, and A. Zettl, Phys. Rev. B59, R2514

~1999!.94P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Phys. Rev. Lett.87,

215502~2001!.95J. Heremans and C. P. Beetz, Jr., Phys. Rev. B32, 1981~1985!.96S. Volz and D. Lemonnier, Phys. Low-Dimens. Semicond. Struct.5–6,

91 ~2000!.97J. Zou and A. Balandin, J. Appl. Phys.89, 2932~2001!.98M. Doi and S. F. Edwards,The Theory of Polymer Dynamics~Oxford,

New York, 1986!.99F. P. Wyart and P. G. de Gennes, Phys. Low-Dimens. Semicond. Stru1,

93 ~2000!.100K. Kurabayashi and K. E. Goodson, J. Appl. Phys.86, 1925~1999!.101International Technology Roadmap for Semiconductors, 1998 Upd

~Semiconductor Industry Association, San Jose, CA, 1999!.102Y. L. Shen, J. Vac. Sci. Technol. B17, 2115~1999!.103B. C. Daly, G. A. Antonelli, H. J. Maris, W. K. Ford, L. Wong, and E

Andideh, Physica B316–317, 254 ~2002!.104W. S. Capinski and H. J. Maris, Rev. Sci. Instrum.67, 2720~1996!.105E. B. Varner, T. Marieb, A. S. Mack, J. Lee, W. K. Meyer, and K.

Goodson, Mater. Res. Soc. Symp. Proc.473, 279 ~1997!.106A. Grill and V. J. Patel, J. Appl. Phys.85, 3314~1999!.107B. K. Hwang, M. J. Loboda, G. A. Cerny, R. F. Schneider, J. A. Seiffer

and T. Washer,Proceedings of the IEEE 2000 International InterconneTechnology Conference~IEEE, Piscataway, NJ, 2000!, pp. 52–54.

108D. G. Cahill, S. K. Watson, and R. O. Pohl, Phys. Rev. B46, 6131~1992!.

109C. Hu, M. Morgen, P. S. Ho, A. Jain, W. N. Gill, J. L. Plawsky, and P.Wayner, Appl. Phys. Lett.77, 145 ~2000!.

110R. M. Costescu, A. J. Bullen, G. Matamis, K. E. O’Hara, and D.Cahill, Phys. Rev. B56, 094205~2002!.

111D. G. Cahill, M. Katiyar, and J. R. Abelson, Phys. Rev. B50, 6077~1994!.

112R. Landauer, inElectrical Transport and Optical Properties of Inhomogeneous Media, edited by J. C. Garland and D. B. Tanner~AmericanInstitute of Physics, New York, 1978!, pp. 2–43.

113J. G. Berryman, J. Acoust. Soc. Am.91, 551 ~1992!.114T. Borca-Tasciuc, A. R. Kumar, and G. Chen, Rev. Sci. Instrum.72, 2139

~2001!.115K. An, K. S. Ravichandran, R. E. Dutton, and S. L. Semiatin, J. A

Ceram. Soc.82, 399 ~1999!.116D. G. Cahill and S.-M. Lee, Microscale Thermophys. Eng.1, 47 ~1997!.


r-

W.

.

et

G.M

T

ty,

.

so

S

ty,

C

D

.

ls

., A

ini,

in.

pl.

g,

st.

ett.

,


117S.-M. Lee, G. Matamis, D. G. Cahill, and W. P. Allen, Microscale Themophys. Eng.2, 31 ~1998!.

118G. Soyez, J. A. Eastman, L. J. Thompson, G.-R. Bai, P. M. Baldo, A.McCormick, R. J. DiMelfi, A. A. Elmustafa, M. F. Tambwe, and D. SStone, Appl. Phys. Lett.77, 1155~2000!.

119V. Narayanamurti, H. L. Sto¨rmer, M. A. Chin, A. C. Gossard, and WWiegmann, Phys. Rev. Lett.43, 2012~1979!.

120M. V. Simkin and G. D. Mahan, Phys. Rev. Lett.84, 927 ~2000!.121S. Y. Ren and J. D. Dow, Phys. Rev. B25, 3750~1982!.122S.-M. Lee, D. G. Cahill, and R. Venkatasubramanian, Appl. Phys. L

70, 2957~1997!.123T. Borca-Tasciuc, W. Liu, J. Liu, T. Zeng, D. W. Song, C. D. Moore,

Chen, K. L. Wang, M. Goorsky, T. Redetic, R. Gronsky, T. Koga, andS. Dresselhaus, Superlattices Microstruct.28, 199 ~2000!.

124T. Yao, Appl. Phys. Lett.51, 1798~1987!.125X. Y. Yu, G. Chen, A. Verma, and J. S. Smith, Appl. Phys. Lett.67, 3554

~1995!.126W. S. Capinski, H. J. Maris, M. Cardona, D. S. Katzer, K. Ploog, and

Ruf, Physica263–264, 530 ~1999!.127S. T. Huxtable, A. R. Abramson, A. Majumdar, C. L. Tien, C. LaBoun

X. Fan, G. Zeng, J. E. Bowers, A. Shakouri, and E. T. Croke, inProceed-ings of the ASME IMECE~ASME, New York, 2001!.

128T. Borca-Tasciuc, D. Achimov, W. L. Liu, G. Chen, H.-W. Ren, C.-HLin, and S. S. Pei, Microscale Thermophys. Eng.5, 225 ~2001!.

129M. A. Afromowitz, J. Appl. Phys.44, 1292~1973!.130G. Mahan, B. Sales, and J. Sharp, Phys. Today50, 42 ~1997!.131R. Venkatasubramanian, Phys. Rev. B61, 3091~2000!.132M. N. Touzelbaev, P. Zhou, R. Venkatasubramanian, and K. E. Good

J. Appl. Phys.90, 763 ~2001!.133W. S. Capinski, H. J. Maris, T. Ruf, M. Cardona, K. Ploog, and D.

Katzer, Phys. Rev. B59, 8105~1999!.134S. T. Huxtable, A. R. Abramson, A. Majumdar, C. L. Tien, C. LaBoun

X. Fan, G. Zeng, J. E. Bowers, A. Shakouri, and E. T. Croke~unpub-lished!.

135S. T. Huxtable, A. R. Abramson, A. Majumdar, C. L. Tien, P. Abraham,LaBounty, X. Fan, Y.-J. Chen, J. E. Bowers, and A. Shakouri~unpub-lished!.

136P. Hyldgaard and G. D. Mahan, Phys. Rev. B56, 10754~1997!.137G. Chen and M. Neagu, Appl. Phys. Lett.71, 2761~1997!.138G. Chen, Phys. Rev. B57, 14958~1998!.139G. Chen, T. Borca-Tasciuc, B. Yang, D. Song, W. Liu, T. Zeng, and

Achimov, Therm. Sci. Eng.7, 43 ~1999!.140S. Tamura, Y. Tanaka, and H. J. Maris, Phys. Rev. B60, 2627~1999!.141G. Chen, J. Heat Transfer121, 945 ~1999!.142W. E. Bies, R. J. Radtke, and H. Ehrenreich, J. Appl. Phys.88, 1498

~2000!.143A. A. Kiselev, K. W. Kim, and M. A. Stroscio, Phys. Rev. B62, 6896

~2000!.144B. Yang and G. Chen, Phys. Low-Dimens. Structures5–6, 37 ~2000!.145S. Volz, J. B. Saulnier, G. Chen, and P. Beauchamp, Microelectron

9–10, 815 ~2000!.


.

t.

.

.

n,

.

.

.

J.

146M. Bartkowiak and G. D. Mahan, inSemiconductors and Semimeta~Academic, New York, 2001!, Chap. 8.

147A. Khitun, A. Balandin, J. L. Liu, and K. L. Wang, J. Appl. Phys.88, 696~2000!.

148A. R. Abramson, C. L. Tien, and A. Majumdar, J. Heat Transfer124, 963~2002!.

149B. C. Daly and H. J. Maris, Physica B316–317, 247 ~2002!.150P. Hyldgaard and G. D. Mahan, inThermal Conductivity~Technomic,

Lancaster, PA, 1996!, p. 172.151H. Kato, S. Tamura, and H. J. Maris, Phys. Rev. B53, 7884~1996!.152G. Chen, T. Zeng, T. Borca-Tasciuc, and D. Song, Mater. Sci. Eng

292, 155 ~2000!.153B. C. Daly, H. J. Maris, S. Tamura, and K. Imamura~unpublished!.154A. Asenov, R. Balasubramaniam, A. R. Brown, J. H. Davies, and S. Sa

IEDM Tech. Dig.2000, 279 ~2000!.155D. A. Young, C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc,

Phonon Scattering in Condensed Matter, edited by A. C. Anderson and JP. Wolfe ~Springer, Berlin, 1986!, p. 49.

156C. A. Paddock and G. L. Eesley, J. Appl. Phys.60, 285 ~1986!.157O. W. Kading, H. Skurk, and K. E. Goodson, Appl. Phys. Lett.65, 1629

~1994!.158B. M. Clemens, G. L. Eesley, and C. A. Paddock, Phys. Rev. B37, 1085

~1988!.159N. Taketoshi, T. Baba, and A. Ono, Jpn. J. Appl. Phys., Part 238, L1268

~1999!.160L. Shi, S. Plyasunov, A. Bachtold, P. L. McEuen, and A. Majumdar, Ap

Phys. Lett.77, 4295~2000!.161C. C. Williams and H. K. Wickramasinghe, Appl. Phys. Lett.49, 1587

~1986!.162A. Majumdar, Annu. Rev. Mater. Sci.29, 505 ~1999!.163G. Binnig, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett.56, 930 ~1986!.164K. Luo, R. W. Herrick, A. Majumdar, and P. Petroff, Appl. Phys. Lett.71,

1604 ~1997!.165A. Hammiche, D. J. Hourston, H. M. Pollock, M. Reading, and M. Son

J. Vac. Sci. Technol. B14, 1486~1996!.166L. Shi, O. Kwon, A. Miner, and A. Majumdar, J. Microelectromech. Sy

10, 370 ~2001!.167L. Shi and A. Majumdar, J. Heat Transfer124, 329 ~2002!.168L. D. Bell and W. J. Kaiser, Annu. Rev. Mater. Sci.26, 189 ~1996!.169G. O. Smith, T. Juhasz, W. E. Bron, and Y. B. Levinson, Phys. Rev. L

68, 2366~1992!.170R. Merlin, Solid State Commun.102, 207 ~1997!.171D. J. Dieleman, A. F. Koenderink, M. G. A. Van Veghel, A. F. M. Arts

and H. W. de Wijn, Phys. Rev. B64, 174304~2001!.172J. J. Baumberg, D. A. Williams, and K. Ko¨hler, Phys. Rev. Lett.78, 3358

~1997!.173H.-Y. Hao and H. J. Maris, Phys. Rev. Lett.84, 5556~2000!.174A. Macovski,Medical Imaging Systems~Prentice Hall, Englewood Cliffs,

NJ, 1983!.


applied physics reviews nanoscale thermal … physics reviews nanoscale thermal transport david g....

Documents