Thermal Transport of Nanoscale Devices

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Downloaded Fstrongly reduced for layer thickness below 100 nm. The through-thickness thermal con-ductivity of Si-Ge and GaAs-AlAs superlattices has recently been shown to be smallerthan the conductivity of the corresponding alloy. The 3v method has been recentlyextended to measurements of anisotropic conduction in polyimide and superlattices.Data for carbon nanotubes measured using micromachined and suspended heaters andthermometers indicate a conductivity near room temperature greater than diamond.@DOI: 10.1115/1.1454111#

Keywords: Conduction, Heat Transfer, Interface, Measurement Techniques, Microscale,Nanoscale, Thin Films

1 IntroductionThe last 50 years have witnessed the transformation of solid-

state devices from objects of mere scientific curiosity to being atthe heart of information technology. Electronic and optoelectronicdevices are now so ubiquitous in modern society that their useoften goes unnoticed. One of the main factors that has enabled thisdramatic transformation is the ability to control the flow of chargein semiconductors. While charge flows only in metals and semi-conductors, heat on the other hand flows in all materials. It is,therefore, natural to ask the question: What are the fundamentaland engineering limits on the control of heat flow in solids?

Heat flow in solids occurs predominantly due to two energycarriers, namely, electrons in metals, and lattice vibrations orphonons in insulators and semiconductors. If one has to controltheir flow, one must recognize two fundamental length scaleswavelength and mean free pathassociated with both electronsand phonons. One way to gain greater control of heat flow is todesign and fabricate solid structures that overlap in size with thefundamental length scales. Table 1 lists the typical range of valuesfor the length and time scales of electrons and phonons. It isobvious that size control of solid structures must occur at nanos-cales. Modern manufacturing techniques allow us to fabricate

structures that are confined to 2 ~films!, 1 ~wires!, or 0 ~dots!dimensions with sizes in the 1100 nm range. Hence, it is nowpossible, to some extent, to manipulate heat flow in solids throughthe interplay of confined size with electron/phonon wavelengthsand mean free paths. Although much remains to be understoodabout heat flow in micro/nanostructures, significant progress hasbeen made over the last decade or so. This could not have beenpossible without the development of new measurement techniquesthat allow one to probe heat flow at length and time scales com-parable to those listed in Table 1. The purpose of this paper istwofold: ~i! to provide a critical review of these measurementtechniques and a discussion of future areas of development; and~ii! to present our current understanding of heat flow in micro/nanostructures and identify areas that need further research.

While greater control of heat flow might indeed be possibleusing confined structures, the following question is rather perti-nent: Why is it important and what impact could it have? As thedemands of information technology keep increasing, new engi-neering challenges continuously emerge. For example, the rapidincrease in power density and speed of integrated circuits requiresthe use of unconventional materials. Figure 1~a! shows a micro-graph of a polycrystalline diamond thin film grown by chemicalvapor deposition on a silicon wafer @1#. The grain size and orien-tation, which depend on the details of the nucleation and growthprocesses, influence the thermal conductivities and thereby affectheat removal rate and response speed. Passivation materials inintegrated circuits are often chosen for their ability to reduce elec-

Contributed by the Heat Transfer Division for publication in the JOURNAL OFHEAT TRANSFER. Manuscript received by the Heat Transfer Division July 27, 2001;revision received December 7, 2001. Associate Editor: V. K. Dhir.

Copyright 2002 by ASMEJournal of Heat Transfer APRIL 2002, Vol. 124 223David G. CahillDepartment of Materials Science and

Engineering,and the Frederick Seitz Materials Research

Laboratory,University of Illinois,

Urbana, IL 61801

Kenneth GoodsonDepartment of Mechanical Engineering,

Stanford University,Stanford, CA 94305

Arunava MajumdarDepartment of Mechanical Engineering,

University of California,Berkeley, CA 94720

e-mail: [email protected]

ThermomTransporSolid-StaStructureWe review recent advatime-resolution thermosurements of thermalcopy (SThM) achieveskHz; SThM has been uand multi-wall carbonresolution in thermalterfaces between matematerials spans a relature. Scanning thermosubmicron lateral resoelectronic and micrombeen demonstrated anbelow the diffractiontance thermometry ancrystal films, multilayelayer dielectrics. The rrom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Teetry and Thermalt in Micro/Nanoscalete Devices ands

nces in experimental methods for high spatial-resolution and highmetry, and the application of these and related methods for mea-ransport in low-dimensional structures. Scanning thermal micros-lateral resolutions of 50 nm and a measurement bandwidth of 100sed to characterize differences in energy dissipation in single-wallnanotubes. Picosecond thermoreflectance enables ultrahigh time-

diffusion experiments and characterization of heat flow across in-rials; the thermal conductance G of interfaces between dissimilartively small range, 20,G,200 MW m22 K21 near room tempera-reflectance microscopy provides nanosecond time resolution andlution needed for studies of heat transfer in microelectronic, opto-echanical systems. A fully-micromachined solid immersion lens has

achieves thermal-radiation imaging with lateral resolution at farimit, ,2 mm. Microfabricated metal bridges using electrical resis-

joule heating give precise data for thermal conductivity of singler thin films, epitaxial superlattices, polycrystalline films, and inter-oom temperature thermal conductivity of single crystal films of Si isrms of Use: http://www.asme.org/about-asme/terms-of-use

in the 5100 nm range. The maximum optical power output ofthese lasers depends critically on the heat removal rate. More

Table 1 Typical length and time scales for energy carriers insolids at room temperature

Downloaded Ftromagnetic cross-talk between neighboring interconnects and de-vices. However, they often have very low thermal conductivities.The importance of thermal properties of passivation materials inintegrated circuits is illustrated by the metal-silicon contact in Fig.1~b!, whose temperature rise during rapid current transients isstrongly influenced by heat diffusion into the bounding silicondioxide @2#. One of the major breakthroughs in fiber optic com-munication is that of dense wavelength division multiplexing~DWDM! @3# in which a single optical fiber carries encoded in-formation from about 100 lasers operating nominally at 1550 nmwavelength but with their wavelengths separated by about 0.40.8 nm. Figure 2 shows the cross-sectional view @4# of vertical-cavity surface-emitting laser ~VCSEL! @5,6#, which is made up ofmultiple layers of IIIV semiconducting materials nanostructured

Fig. 1 Electron micrographs of novel electronic films and de-vices: a Diamond passivation film on aluminum. Diamondfilms are promising for enhanced heat removal from power in-tegrated circuits and for fast thermal sensors. b Transmissionelectron micrograph of a VLSI metal-silicon contact, which hasfailed during a current pulse of sub-microsecond duration. Thetemperature rise is strongly influenced by the properties of thesurrounding passive material.

224 Vol. 124, APRIL 2002rom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Teimportantly for DWDM, because the refractive index of the IIIVmaterials is temperature dependent, the output wavelength varieswith temperature at about 0.1 nm/K, which can lead to cross-talkbetween channels. Hence, it is necessary to actively control theirtemperature, which requires one to develop solid-state coolingdevices. Figure 3 shows an array of thermionic microcoolers fab-ricated on a silicon wafer @7#. The cross-section of this deviceshows that it contains a Si/SiGe superlattice with a period of about10 nm. The purpose of this superlattice is to minimize heat flowbetween the cold and hot ends of the refrigerator. This leads to thequestion as to why such solid-state refrigerators and engines can-not replace their macroscopic vapor based ones.

It is often forgotten that when semiconductors were first discov-ered, the initial excitement about their use was not for electronicsbut for refrigeration @8#. The reason we do not generally use solid-state energy conversion is because of their low device perfor-mance. This depends on the non-dimensional thermoelectric fig-ure of merit of a material, ZT5S2sT/k , where S is the Seebeckcoefficient, s is the electrical conductivity, and k is the thermalconductivity. Currently, compounds such as Bi2Te3 offer the bestZT (ZT1) at room temperature, which leads to coefficient ofperformance of refrigerators of about unity. If ZT could be in-creased to 3, the performance of thermoelectric refrigerators orpower generators could be comparable to their gas-based counter-parts. Although it is possible to control S and s by doping asemiconductor, significant reduction of k has not been possible inthe past. Recent studies have shown that by nanostructuring asolid, it would be possible to simultaneously increase S2s andreduce k and thereby increase its ZT value to 3 or more @9#. This,by itself, is a compelling reason to study heat flow in micro/nanostructured solids.

The paper is divided into two main sections, one on thermom-etry at short length and time scales and the other on thermal trans-port in low-dimensional structures.

2 ThermometryDuring the last decade, remarkable progress has been made in

the area of microscale thermometry, which can now allow tem-perature measurements at length and time scales comparable tothe mean free paths and relaxation times of energy carriers insolids ~see Table 1!. This section provides a critical review ofthree thermometry techniques, namely: ~i! scanning thermal mi-croscopy for high spatial resolution thermal imaging; ~ii! picosec-ond reflectance thermometry for measuring thermal properties ofinterfaces and nanostructures; and ~iii! scanning optical thermom-etry that combines high spatial and temporal resolution measure-ments.

2.1 Scanning Thermal Microscopy. A scanning thermalmicroscope ~SThM! operates by bringing a sharp temperature-sensing tip in close proximity to a sample solid surface ~see Fig.4!. Localized heat transfer between the tip and sample surfacechanges the tip temperature. By scanning the tip across the samplesurface, a spatial distribution of the tip-sample heat transfer ismapped out. When the tip comes in local equilibrium with thesample, one obtains the spatial temperature distribution of samplesurface, whereas if the temperature change in determined for aknown heat flux, one could obtain the local thermal properties.The spatial resolution depends mainly on three factors, namely, tipsharpness, tip-sample heat transfer mechanism, and the thermaldesign of the probe. In the recent past, the spatial resolution hasbeen improved to 3050 nm, allowing one to study thermal trans-port phenomena at these length scales.

The idea of SThM was first proposed by Williams and Wickramasinghe @10# when they used a thermocouple at the end of ametallic wire to measure temperature. Tip-sample heat transferwas used to image the surface topography of electricallyinsulating materials. Since then, significant progress has

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be@1comfirst study utilized a wire probe @10,11#, most of the subsequent

.

-r

The thermal design of the cantilever probes is extremely impor-

J

Downloaded FroFig. 3 Electron micrographs of left an array of SiGe superlattice thermionic microcoolers on a single wafer and right thecross section of a single device showing the SiGeSi superlattice structure. The superlattice period is approximately 10 nm.

ournal of Heat Transfer APRIL 2002, Vol. 124 225development has been towards cantilever-based probes such thatatomic force microscopes ~AFMs! @28# could be used as a plat-form for SThM. Figure 4 shows a schematic diagram of an AFM,which utilizes a sharp tip mounted on a cantilever to be scannedacross a sample surface. Cantilever deflections are measured byreflecting a laser beam off the cantilever and onto a position-sensitive photodiode. The deflection signal is used in a controlsystem to maintain a constant tip-sample contact force while thetip is scanned laterally. When a temperature sensor is mounted onthe very apex of the tip, such probes can be used to image both thetopography and the temperature distribution of devices such assingle transistors @12,13,29# and vertical cavity lasers @30#. Suchstudies have helped in identifying defects and failure mechanismswhile providing insight about electron, photon and phonon trans-port in these devices. In addition, SThM has also been used tomeasure thermal properties of materials and perform calorimetryat nanometer scales @14,16,20,22,23,24,31#. Progress in this fieldm: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Tetant for SThM performance. The thermal resistance network inFig. 4 suggests that for a given ambient temperature, Ta , andsample temperature Ts , the tip temperature Tt , can be written asTt5Ts1(Ta2Ts)/(11f), where f5(Rc1Rt)/Rts is the ratio ofthe cantilever thermal resistance, Rc , plus the tip resistance, Rt ,with respect to the tip-sample thermal resistance, Rts . The canti-lever and tip resistance arises from both conduction through thetip and cantilever material as well as the surrounding air, while thetip-sample resistance results from liquid, air and solid-solid con-duction. Changes in sample temperature can be related to changesin the tip temperature as DTt /DTs5f/(11f). This relation sug-gests that the accuracy and sensitivity of sample temperature mea-surement by the tip depend on f, which must be large for betterSThM performance. The spatial resolution, Dx , of SThM mea-surements can be expressed as Dx5DTn /(dTt /dx) where DTn isthe noise in the temperature measurement and dTt /dx is the mea-en made in improving thermocouple-based measurements1,12,13,14,15,16,17,18,19#, while other techniques based onntact potential @20#, electrical resistance @21,22,23,24# and ther-al expansion @25,26,27# have also been developed. Although the

till 1999 has been documented in a comprehensive review @32#Since then, however, there have been some significant breakthroughs in both probe design and fabrication, and heat transfestudies at nanoscales @33#.

Fig. 2 Electron micrographs of left an array of vertical cavity surface emitting lasers VCSEL on a single wafer and rightthe cross of an individual VCSEL. Each laser contains multiple layers of IIIV semiconductor materials which are approxi-mately 60100 nm in the Bragg mirrors of the laser or 110 nm in the active quantum well region.rms of Use: http://www.asme.org/about-asme/terms-of-use

Downloaded Fsured temperature gradient. Because the tip and sample tempera-tures are related through f, the spatial resolution can be expressedas

Dx5DTn

~dTs /dx !S 11ff D . (1)

Equation ~1! clearly suggests that small values of f lead to poorspatial resolution of SThM. While the contributions of both airand solid conduction to Rc and Rt depends on probe geometry,that for solid conduction depends critically on the thermal conduc-tivity of tip and cantilever material. Shi et al. @34# have performedelaborate modeling of air and solid conduction between thesample and the cantilever probe, including microscale effects nearthe tip-sample contact, to study the effects of material propertiesand probe geometry on Rc and Rt . This has provided a frameworkfor designing SThM probes.

One of the most challenging problems in SThM is probe fabri-cation. Initial studies had focused on etched wire probes@10,11,12,13,14#, although the high thermal conductivity of metalwires as well as irreproducibility and the large sensor size led toinaccuracies and lack of resolution. This led to development ofthin film probes that could be microfabricated. However, toachieve the nanoscale sensor size at the apex of the tip, non-traditional lithography techniques had to be used, which requiredthe probes to be made one at a time @15,17#. Subsequently, batchfabricated probes were developed, first based on e-beam lithogra-phy @19# and later using standard photolithography-based MEMSfabrication techniques @34#. Details of the design process and fab-rication process can be found in Shi et al. @34#. Figure 5 showselectron micrographs of a probe @34# containing two cantileverarms a Pt-Cr thermocouple junction at the apex of the pyramid-shaped tip. The tip radius is generally about 50100 nm while thethermocouple size can be controlled between 100500 nm inheight.

A fundamental understanding of tip-sample heat transfermechanisms is crucial for quantitatively probing nanostructuresusing SThM. Although researchers initially thought heat transferto be dominated by solid-solid and air conduction, the phenom-

Fig. 4 Schematic diagram of a scanning thermal microscopeSThM. It consists of a sharp temperature sensing tip mountedon a cantilever probe. The sample is scanned in the lateral di-rections while the canitilever deflections are monitored using alaser beam-deflection technique. Topographical and thermalimages can be thermally obtained. The thermal transport at thetip-sample contacts consists of air, liquid and solid-solid con-duction pathways. A simple thermal resistance network modelof the sample and probe combination shows that when thesample is at temperature Ts , the tip temperature Tt depends onthe values of the thermal resistances of the tip-sample contact,Rts , the tip, Rt , and the cantilever probe, Rc .

226 Vol. 124, APRIL 2002rom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Teenon turned out to be more complex. One of the surprising dis-coveries was that of the role of liquid film conduction. A solidsurface is generally covered with a very thin layer ~a few mono-layers thick! of a liquid film under ambient conditions: water forhydrophilic surfaces and some hydrocarbon for hydrophobic ones.When the tip comes in contact with the sample, the liquid filmforms a meniscus and bridges the tip and the sample. It was firstproposed by Luo et al. @17# that this liquid film dominated tip-sample heat transfer for nanoscale point contacts, and suggested aheat transfer coefficient h109 W/m2-K. The actual conductancedepends on the radius of curvature of the tip apex. More recently,Shi et al. @35# have shown using the probes in Fig. 5, that whileliquid film does play an important role, solid-solid and air con-duction pathways are equally important. Figure 6 plots the canti-lever deflection and the tip temperature as a function of tip-samplerelative position. The measurements show that before the tip isbrought into contact, air conduction dominates heat transfer. Asthe tip makes contact with the sample surface, liquid film conduc-tion results in a sudden jump in tip temperature due to a markedincrease in thermal conductance. Subsequently, the tip tempera-ture rises gradually as the contact force is increased indicating therole of solid-solid conduction. Beyond a certain force, the tip tem-perature saturates to a constant value, which arises from the factthat the tip-sample contact area becomes constant. From this data,values of liquid, solid-solid and air conductances can be deter-mined @35#. Although this is a typical thermal behavior of SThMprobes, the exact values of forces and thermal conductances atdifferent transition points depends on the probe design. The ther-mal time constant of probes shown in Fig. 5 during contact with asample was shown to be on the order of 10 ms @34#, which canallow a measurement bandwidth of about 100 kHz @36#.

Figure 7 shows topographical and thermal images of the cross-section of a vertical cavity surface emitting laser ~VCSEL! @30#.Because the refractive index of the IIIV semiconducting materi-als used in VCSELs is temperature dependent, the output wave-length of VCSELs varies with temperature at about 0.1 nm/K.Hence, not only is it necessary to actively control the temperatureof these devices @7#, fundamental understanding of thermal phe-nomena inside the devices is also important. SThM images of aVCSEL showed that the maximum temperature occurred at thequantum well region where electrons and holes recombine radia-tively. This suggested non-radiative recombination due to thepresence of defects and crystal imperfections to be the dominantloss mechanism as opposed to other mechanisms such as Jouleheating and radiative absorption. The temperature distribution alsosuggested a thermal lens formation due to the temperature depen-dence of the refractive index of these materials.

Carbon nanotubes have recently been intensively studied fortheir electrical, thermal, mechanical, and chemical properties @37#.They come in two kinds: multiwall carbon nanotubes ~MWCNs!,which are generally about 10 nm in diameter, and single wallcarbon nanotubes ~SWCNs! which are about 12 nm in diameter.

Fig. 5 Electron micrographs of a two-armed cantilever probewith a pyramid-shaped tip at the free end. The tip contains aPt-Cr thermocouple junction at its apex which is approximately500 nm in lateral size.

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While current flow in MWCNs is expected to be diffusive leadingto uniform Joule heating, there is speculation that electron trans-

perature measurements; ~ii! design and fabrication of integrated

Fpwsample approaches the tip, the cantilever deflection remainsunchanged till a jump to contact. During this time, the tip tem-

e

Downloaded FMWCNs are generally metallic in nature, while SWCNs can beboth metallic and semiconducting depending on the chirality intheir crystal structure. Although there have been several studies onelectrical transport in MWCNs and SWCNs there are several un-answered questions regarding dissipation in these nanostructures.

perature rises very gradually due to air conduction. Corre-sponding to jump to contact the tip temperature suddenly risesdue to liquid conduction. As the sample is pushed up the can-tilever deflects linearly, while the tip temperature increases lin-early due to increase in solid-solid contact area to a certainconstant value. When the sample is retracted away, the solid-solid conductance decreases, but at a different rate, indicatingsome hysteresis due to plastic deformation of the tip or thesample. The tip remains in contact for a longer duration due tosurface tension forces. When the tip snaps out of contact fromthe sample, there is a sudden drop in temperature due to lossof liquid film conduction.

Fig. 7 left Schematic diagram, middle topographical imagcavity laser. The region imaged is approximately 6 mm on eithp-doped and n-doped Bragg mirrors consisting of alternatingafter a current was applied and the device was lasing.

Journal of Heat Transferrom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Temultifunctional probes; ~iii! thermal probing of new devices andmicro/nanostructures. Although we have made progress in our un-derstanding of heat transfer at point contacts @17,35,40#, there arestill many fundamental issues unresolved. For example, can theeffect of liquid conduction be controlled by tailoring surfacechemistry? What is the role of near-field radiation in tip-sampleheat transfer @41,42,43#, especially in vacuum environment? Howdo phonons and electrons propagate through a point contact that ison the order of their wavelength? Can we conduct SThM experi-ments at low temperature and observe quantum transport effects?Because tip-sample thermal resistance depends on surface topog-raphy, topography-related artifacts in thermal images still remainsa problem. There are two potential solutions: a thorough knowl-edge of tip-sample heat transfer in order to model the effect oftopography; or a null-point measurement such that tip-sampletemperature difference can be mini mized thereby eliminating anyheat flow. The latter requires more sophisticated probes with mul-tiple sensors and feedback control.

, and right thermal image of the cross-section of a verticaler side of the active quantum well region, which includes thelayers of AlAs and AlGaAs. The thermal image was obtained

APRIL 2002, Vol. 124 227port in SWCNs is ballistic over several micrometers in lengtheven at room temperatures. Figure 8~a! shows topographical andthermal images of an electrically heated MWCN of 10 nm diam-eter @33,38#. The spatial resolution of SThM was about 50 nm.Detailed modeling of the measured temperature combined with anunderstanding of tip-sample heat transfer mechanism was used toquantitatively determine the actual temperature rise of the MWCN@33#. The temperature profile along the length of the MWCN wasparabolic, indicating a uniform Joule heating and diffusive elec-tron transport. Figure 8~b! shows the topographical and thermalimages of a SWCN that was about 1 nm in diameter. Thermalimages at low voltages showed that most of the temperature risewas near the contacts, indicating ballistic electron flow in theSWCN. At high voltages, however, bulk dissipation started to oc-cur as indicated by the thermal images. This is presumably be-cause of optical phonon emission by electrons when they gethighly energized by transport in high electric fields @39#.

There have been several important developments and break-throughs in SThM over the last decade, which are now allowingus to routinely make thermal measurements at spatial resolutionsin the sub-100 nm regime. There are three main areas whereprogress could be made in the future, namely: ~i! fundamentalunderstanding of tip-sample heat transfer and accuracy of tem-

ig. 6 Cantilever deflection and temperature response of therobe as a function of sample vertical position. The sampleas a 350 nm wide electrically heated metal line. When therms of Use: http://www.asme.org/about-asme/terms-of-use

changes and phase transitions at different temperatures. Localizedcalorimetof biomo

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Downloaded From: httpry of molecules could shed light on the binding behaviorlecules.

icosecond Reflectance Thermometry. Reflectanceetry can be extended to picosecond time-scales usingked lasers; mode-locked laser typically produce a seriespules at a repetition rate of 76 MHz. Extremely high

lution is achieved by splitting the laser output into twohs, a pump beam and a probe beam, and adjustingve optical path lengths with a mechanical delay stage. Ifle under investigation is metallic, or can be coated with aal film, then a small fraction of the energy from eachthe pump beam produces a sudden jump in temperaturesurface of the sample. The decay of the near-surfacere is then interrogated by the reflected energy of thepulses in the probe beam. Typically, the intensity of theam is modulated and variations in the intensity of theprobe beam at the modulation frequency are measuredf lock-in-amplifier.6, Paddock and Eesley @47# described the first thermalexperiments using picosecond thermoreflectance; their

motivationproviding a direct method for measuringsion on nanometer length scalesremains valid today.ineering materials have thermal diffusivities D in the

interpret quantitatively because ~i! hot-electrons can depergy outside of the optical absorption depth, ~ii! the temdependence of the complex index of refraction dn/dT isunknown, ~iii! the diffusion equation fails on length scalparable to the mean-free-paths of the dominant energyand ~iv! the assumption of equilibrium between phonons atrons is not always valid. At long times t.500 ps, measuare often plagued by shifts in the focal plane and beamcreated by large displacements of the delay stage. ~CapiMaris describe a method for improving accuracy at lontimes @56#.! We are optimistic, however, that improvedmance and dropping costs of solid-state mode-locked laexpand applications of picosecond thermoreflectance. Rlow-noise Ti:sapphire lasers that produce 100 fs pulses amercially available and can be operated with only aknowledge of the technology of ultra-fast lasers.

An example apparatus in use at the Laser Facility of terick Seitz Materials Research Laboratory is shown in Figoptical design contains some unique features: a single micobjective focuses the pump and probe beams, collimateflected probe beam, and forms a dark-field microscopy ithe sample surface on a CCD camera. The integrated doptical microscope greatly facilitates focusing and align

ol. 124, APRIL 2002 Transactions of theThe development of batch fabrication processes for making SThMprobes @19,34,44,45# has made them readily available to manyresearchers, which itself is a significant engineering contribution.Currently, however, one can place only a single temperature sen-sor on SThM tips. Future progress will be driven towards multiplesensors and actuators on integrated cantilever probes such that onecould simultaneously measure temperature and heat flux as wellas electrical, optical and other nanoscale properties while imaginga nanostructure. Finally, because new nanoscale materials and en-ergy conversion devices are being developed, there will also re-main a need to thermal probe them at nanoscales. For example,thermoelectric properties of nanostructured semiconductors aremarkedly different from their bulk properties @46#. By simulta-neously measuring thermal, electrical, and thermoelectric proper-ties using a multifunctional SThM, one could characterize low-dimensional nanostructures. An area of research that is largelyunexplored is SThM under liquid environment, in particular, ofbiological molecules. Proteins and DNA undergo structural

Fig. 8 Topographical and thermal images of electriabout 10 nm in diameter with 22.4 mA flowing throughdiameter with 15 mA top and 19.3 mA flowing throughnature indicating a diffusive flow of electrons. At low bing ballistic flow, while at higher bias the SWCN is hemission. All measurements were made at room temp://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Terange 0.005,D,1 cm2 s21 and at t5200 ps, heat diffusionlengths l 5A(Dt) are in the range 10,1,140 nm. Therefore,picosecond time resolution offers nanometer-scale depth resolu-tion and, more specifically, picosecond thermoreflectance can iso-late the effects of interface conductance from the thermal conduc-tivity of a thin layer. By contrast, thermoreflectance usingQ-switched lasers @48# and the 3v method @49# cannot distinguishbetween the thermal conductivity of a film and the thermal con-ductance of its interfaces. Greater understanding of energy trans-port on picosecond time-scales could also enhance capabilities forthin-film metrology. Picosecond acoustics using pump-probe op-tical techniques is a powerful tool for determining film thickness;in some cases, picosecond thermoreflectance could providecomplementary data on film and interface microstructure @50#.

Despite the promise of these methods, only a few quantitativeand systematic studies of thermal properties of materials haveused picosecond optical techniques @47,51,52,53,54,55#. Experi-ments by Maris and co-workers on interface thermal conductance@52# and superlattice conductivity @54# are the most complete.

ally heated: a multiwall carbon nanotube MWCNit; b single wall carbon nanotube SWCN 1-2 nm init. The temperature profile of the MWCN is parabolic inias, the SWCN is heated only at the contacts suggest-eated in the bulk, most likely due to optical phononrature.rms of Use: http://www.asme.org/about-asme/terms-of-use

Downloaded Fthe pump and probe beams. The 1/e2 beam diameter at the sampleis relatively large, 15 mm for both pump and probe.

In a modulated pump-probe experiment, the differences in re-flected probe intensity caused by the pump pulse appear at themodulation frequency of the pump beam and are extracted withlock-in detection. But interpretation of this lock-in signal is notcompletely straightforward, particularly as the time delay be-comes a significant fraction of the separation between pulses. Fol-lowing Ref. @56#, the lock-in response VLI of a modulated pump-probe reflectivity measurement is

VLI~ t !5a

t (q52

exp~ i2p f ~qt1t !!DR1~qt1t !, (2)

where the real part of VLI is the in-phase signal of the lock-inamplifier V in and the imaginary part of VLI is the out-of-phasesignal Vout ; a is a constant, t the time between pulses (typically13 ns), f the modulation frequency, t the time delay of the probebeam, and DR1(t) is the reflectivity change at a time t followingan excitation of the sample by a single pump pulse. In Ref. @56#,the authors state that Eq. 2 is valid only in the limit f t!1 but wehave found that Eq. 1 is an exact solution for all values of f t aslong as the lock-in time constant tLI satisfies f tLI@1. Typically,comparisons of experimental data and the prediction of Eq. 2 areused to optimize free parameters in a thermal model; for all butthe shortest times, the temperature is homogeneous through theoptical absorption depth of the thin metal film and DR1(t)5(dR/dT)DT1(t). The surface temperature DT1(t) can be accu-rately calculated using one-dimensional heat flow since the ther-mal diffusion length is usually much smaller than the radius of thefocused pump beam @56#.

Although Bonello and co-workers @57# discuss the in-phase sig-nal at t,0, to the best of knowledge, no prior work has made useVout(t). But the extra information in the out-of-phase signal mayprovide a simple method of correcting for non-idealities in theexperiment: both the in-phase and out-of-phase signals will bechanged by the same amount by defocusing of the pump beamand changes in the pump-probe overlap and so the ratio ofV in /Vout is often a more robust quantity than V in alone. Vout(t)varies weakly with t but decreases with modulation frequency f;

Fig. 9 Picosecond thermoreflectance and acoustics appara-tus at the University of Illinois using a single objective and anintegrated dark-field microscope. The pump and probe beamsare parallel at the back-focal-plane of the objective lens butoffset by 4 mm. The aperture in front of the detector rejectsthe small fraction of pump beam that leaks through the polar-izing beam splitter.

Journal of Heat Transferrom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Tewe choose f 10 MHz to minimize radial heat flow without sac-rificing too much signal in Vout .

A frequency domain version of Eq. 2 is sometimes more con-venient: instead of starting with the time domain response to apump pulse DT1(t), we calculate the frequency domain responseDT(n) of a multilayer sample using matrix methods @58#. Withthe probe beam delayed by a time t, the ratio of the two lock-insignals is then

Vin~ t !iVout~ t !

5

(q52

~DT~q/t1 f !1DT~q/t2 f !!exp~ i2ptq/t!

(q52

~DT~q/t1 f !2DT~q/t2 f !!exp~ i2ptq/t!.

(3)If the pump beam is advanced instead of delaying the probe, ~seeFig. 9!, Eq. 3 must be corrected by a phase factor that is equiva-lent to shifting the phase of the lock-in reference channel by2p f t .

Figure 10 shows an example of this approach using data for anepitaxial TiN layer deposited on MgO by reactive magnetron sput-tering at 800C. Because TiN and MgO have the same crystalstructure ~fcc NaCl! and almost identical lattice constants, we ex-pect that these TiN/MgO interfaces are more perfect than typicalmetal/ceramic interfaces. TiN is a metallic ceramic, a hard,refractory material with metallic electrical properties. Our thermalmodel assumes that the heating produced of the pump pulse canbe approximated by an instantaneous deposition of energy uni-formly distributed through the optical absorption depth of TiN,l/(4pk)516 nm. Since we assume DR(t)5(dR/dT)DT(t), the

Fig. 10 Picosecond thermoreflectance data solid circles forthermal transport through an epitaxial TiNMgO001 structureat room temperature. The ratio of the in-phase to out-of-phasesignals of the lock-in amplifier at f9.8 MHz is plotted as func-tion of the thermal diffusion length in TiN; the delay time t isgiven on the top axis. The TiN layer thickness is 100 nm. Weassume that the thermal conductivity of the TiN film is given bythe Wiedemann-Franz law, LLTr58 W m1 K1; the ther-mal conductivity of MgO is 51 W m1 K1. The solid and dashedlines are fits to the data using Eq. 3 with one free parameter, theinterface thermal conductance G. The solid line is the best fit,G0.42 GW m2 K1. The upper and lower dashed lines are forG0.2 and G1.0 GW m2 K1, respectively.

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fit at short times is poor; i.e., we have not accounted for gradientsin n within the optical absorption depth @59#. At long times, we

probabilities for electron interband transitions @78,79#. Becausethe thermoreflectance coefficient can depend on the radiation spot

Downloaded Fobtain an excellent one-parameter fit to the data but the data arerelatively insensitive to the property we are interested in, the ther-mal conductance of the interface G: changing G by a factor of 2produces only a 15 percent change in V in /Vout at t51 ns.

2.3 Scanning Optical Thermometry. The optical ther-mometry described in section 2.2 uses brief laser pulses to exciteand interrogate a surface. The radiation is generally not stronglyfocused, which allows one-dimensional transport modeling nor-mal to the absorbing surface. For micro and nanoscale semicon-ductor devices, sensors, and actuators, the goal is to measure tem-perature distributions resulting from electrical heating withsimultaneously high spatial and temporal resolution. In contrast tothe scanning probe methods described in section 2.1, most opticalthermometry methods directly probe the microdevice without re-quiring heat diffusion into a solid sensor. This eliminates the as-sociated delay and, in principle, allows measurements in the pico-second regime as discussed in section 2.2. Pump-probemicrodevice thermometry with picosecond temporal resolutioncould be achieved using a photodiode to convert the pump pulseinto an electrical current. This electro-optic excitation approach iswell established for measuring electric field distributions and thevoltage step response in semiconductor devices @60,61#. However,the fastest optical thermometry of electrically-heated devices usedsteady-state radiation and yielded temporal resolution, limited bynoise and calibration complications, near 1 ns.

There has been extensive research on microdevice optical ther-mometry. Most measurements use visible or near-visible radiationwith far-field optics, which provide diffraction-limited spatialresolution near 1 mm @62#. Examples include micro-Raman ther-mal imaging, for which the ratio of the Stokes and anti-Stokeslines determined the steady-state optical phonon temperature inthe semiconducting regions of quantum well lasers @63# and sili-con field-effect transistors @64#. Picosecond Raman thermometryof solvent-solute mixtures at large spatial dimensions @65# couldbe extended to microdevices. Expansion thermometry @66,67,68#used radiation interference or dilation to probe thermal expansionand temperature fields in the near-surface region. The extractionof temperature fields from surface displacement is very difficult inthe absence of a dedicated surface layer with high thermal expan-sion coefficient, as discussed for Joule expansion microscopy us-ing the AFM in section 2.1. The temperature dependence ofphoton-induced emission provides another opportunity for ther-mometry, although the governing electron-photon relaxation pro-cesses can limit temporal resolution. Example studies usedtemperature-induced shifts in the photoluminescence spectra ofsemiconductors @69# or, for the case of a dedicated thin-film coat-ing, a shift in the fluorescence efficiency @70#.

Thermoreflectance imaging detects temperature changesthrough the optical reflectance @71# and has been applied to mi-cromachined structures @72,73,74,75#. Precise scanning of the la-ser focus spot over the microdevice during periodic electricalheating yields the full-field transient temperature @76#. Figure11~a,b! show the experimental setup and temperature distributiondata across the corner of an aluminum-copper alloy interconnect@77#. The temperature distribution was captured 100 ns after theinitiation of an electrical current pulse and is consistent with acoupled analysis of the temperature and potential distributions inthe metal. The diffraction-limited spatial resolution is comparablewith the radiation wavelength of 800 nm, which was chosen tooptimize the thermoreflectance coefficient, CTR . Although metalshave relatively low thermoreflectance coefficients, CTR;1025 K21, they are well suited for thermoreflectance imagingbecause their small penetration depths isolate the volume overwhich radiation-surface interactions are occurring and minimizesthe impact on the electrical behavior of the device. The thermore-flectance coefficients of aluminum and copper achieve maximanear 800 nm and 577 nm, respectively, owing to associated high

230 Vol. 124, APRIL 2002rom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Tesize, a two-step calibration procedure on a dedicated, similar mi-cromachined structure subjected to transient electrical heating andthermometry is helpful for obtaining the relevant value of CTR@76#. Reflectance thermometry can be performed through transpar-ent passive layers, such as silicon dioxide. However, temperatureand refractive index gradients in the overlayer effectively limit thetemporal resolution to the thermal diffusion time normal to thatlayer.

Optical thermometry below the diffraction limit remains imma-ture. From the far field, radiation with vacuum wavelength l canbe focused to a spot whose full width at half maximum is l/~2NA!. The numerical aperture of the scanning system is NA5n sin u, the maximum angle of incidence is u, and n is the realpart of the refractive index. This is equivalent to Sparrows crite-rion, in which diffraction limits the spatial resolution in a vacuumor air (n51) to l/2. Optical thermometry below the diffractionlimit is based on near-field scanning optical microscopy ~NSOM!,which achieved resolution approaching 10 nm for optical imagingusing a probe scanned a few nanometers above the surface. OneNSOM method uses tapered, metal-coated optical fibers with anaperture at the tip @80#. Near-field infrared imaging used a taperedmetal micropipette to collect infrared radiation emitted by a pat-terned metal bridge structure @81#. The fiber aperture was near 1.5mm, which is almost an order of magnitude less than the dominantemitted wavelength at room temperature. Another technique de-livered visible light through a metal-coated fiber with aperturenear 50 nm @82#. The radiation reflected into the far field was usedto image temperature differences between two metal bridges. Fortransient measurements, data interpretation was complicated bythermal expansion of the surface and probe. These exploratoryscanning probe studies showed that the low transmittance throughapertured probes (;1026) can strongly limit the sensitivity fortemperature measurements.

These problems may eventually be mitigated using solid im-mersion lens ~SIL! technology, which improves spatial resolutionas in oil immersion microscopy. Figure 12~a! indicates that a solidlens scanning in the near field can provide spatial resolution belowthe wavelength in air without the dramatic loss in transmissionimposed by an apertured fiber. The plotted sixth-power depen-dence of transmission is based on the result of Bethe for apertures@83#. The SIL increases the numerical aperture of the optical pathaccording to the real part of the refractive index of the lens ma-terial. The SIL with visible or ultraviolet radiation has achievedvalues of NA between 1.0 and 2.2 for optical imaging @84#, datastorage @85#, and photolithography @86#. In principle, SILs withtailored optical properties should be appropriate for improving thespatial resolution of the various optical thermometry techniquesdescribed already in this section.

The first fully-micromachined silicon SIL @87# targeted nearand mid-infrared radiation wavelengths, for which silicon isnearly transmitting with n;3.4. Figure 12~b! shows the integratedsilicon lens and cantilever. Lenses were fabricated by developingphotoresist pillars on the silicon film on a silicon-on-insulator~SOI! substrate. The pillars were shaped into hemispheres duringexposure to acetone vapor, and the shape was transferred into thesilicon film using CF4 /O2 reactive ion etching. Using transmis-sion through a 1 mm aperture patterned in Cr/Au on a GaP sub-strate, the lens produced a spot size of l/5 and NA52.5 withradiation wavelength l59.3 mm. Subsequent research imagedthermal radiation transmission through a micropatterned intercon-nect structure on silicon @88#. Figure 12~c! shows the improvedresolution of the radiance images collected during a scan acrosstwo aluminum lines of width 1 mm and separated by 2.5 mm.Figure 12~c! is not a thermal map of the interconnect structure, butrather an image of the thermal radiation transmitted through themicrostructure from an underlying heater, which shows that localmeasurements of temperature gradients should be possible. A

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noise equivalent tresolution of 1.7 mlimited infrared mat 20 Hz. Althougmay be improvedthe lens, the resunoise-limited temtractive for therm

3 Thermal Tr

3.1 Thin Filmstrongly influenceeffects. The discrtivities can in manintroduced duringof thickness compboundary scatterinity. Additional detity reviews @89,9

lms @95,96#, andods @100#.

thermal conduc-by the source and. Many use elec-erned above theurrent in a metalions at 2v of there amplitude andmaterials are ex-along the bridge.lane thermal con-extensions usingthermal conduc-e @105#, and theMeasurements atlumetric heat ca-e electrical heat-an alternative for4# and the out-of-

Journal of Heat Vol. 124 231

Downloaded From: http://heattraemperature resolution below 0.2 K and a spatialm were predicted at 120C using a diffraction-icroscope with InSb focal plane array detectorh the spatial resolution for optical microscopythrough the use of a metal aperture patterned onlting reduction in optical transmittance and theperature sensitivity makes this approach unat-al imaging.

ansport in Low-Dimensional Structures

s. Thermal conduction in solid films can bed by material imperfections and sub-continuumepancy between bulk and film thermal conduc-y cases be traced to imperfections or impuritiesthe fabrication process. For single-crystal filmsarable with the phonon mean free path, phonon-g also reduces the effective thermal conductiv-ail can be found in thin-film thermal conductiv-0# and more focused articles on superlattices

@91,92,93#, interface resistance @94#, diamond fiorganic films @97,98,99#, and experimental meth

3.1.1 Measurement Techniques. Thin filmtivity measurement techniques are distinguishedtimescale of heating and the thermometry methodtrical heating and thermometry in bridges pattsample film. The 3v method @101# sustains a cbridge at angular frequency v, inducing fluctuatheating rate and the temperature. The temperatuphase and the thermal properties of surroundingtracted from the 3v component of the voltageWhile the 3v method first measured the out-of-pductivity in dielectric films @102,103#, recentmultiple bridges of varying width measured thetivity anisotropy in superlattices @104#, polyimidin-plane conductivity of thin silicon films @106#.heating frequencies up to 60 kHz yielded the vopacity of silicon dioxide films @107#. Steady-stating and thermometry in parallel metal bridges isdetermining film-substrate interface resistance @9

Fig. 11 a Schematic of the setup for scanning laser-reflectance imaging of interconnectsand semiconductor devices; b experimental data for the temperature distribution acrossthe corner of an electrically heated metal intercorrect. For details see Refs. 62,76,77.

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Downloaded Fplane thermal conductivities of disordered dielectric films @108#.Steady and transient heating in metal were used in membranestructures, generally fabricated using anisotropic chemical etch-ing, to isolate the in-plane thermal conductivity @109,110,97#.

Pulsed laser heating and thermoreflectance thermometry havebeen used for non-contact measurements of the out-of-plane con-ductivity. The instrumentation and analysis depends on the heatingpulse duration. As discussed in section 2.2, measurements withpicosecond-scale heating can examine transport in a variety ofthin films and nanostructures. These measurements use pump-probe laser diagnostic to capture both the thermoreflectance re-sponse and the surface displacement resulting from acousticwaves. Analysis needs to consider the non-diffusive, sub-continuum nature of heat transport in the sample films and thedisequilibrium between electrons, which can absorb much of the

radiation, and the lattice. Nanosecond heating from a Nd:YAGlaser and continuous time-domain laser-reflectance thermometryyielded the out-of-plane conductivities of silicon-dioxide @111#and polycrystalline diamond @112# films of thickness down to afew hundred nanometers. The lateral thermal conductivity of filmson substrates has been measured using the transient thermal grat-ing technique @113#. Pulsed laser radiation interferes on thesample surface, yielding a harmonic spatial variation of energyabsorption. This yields spatially varying temperature and thermalstrain fields, whose temporal decay is detected by the deflection ofan incident probe beam. The temporal decay is governed mainlyby the lateral thermal conductivity of the film and substrate andthe volumetric heat capacity within a depth near the spatial periodof energy deposition, which can be varied by altering the differ-ence in incident angles between the two probes.

Fig. 12 a Relationship of the transmittance and spatial resolution of optical imaging technologies, showing the promise of thesolid immersion lens for thermal imaging. The curve plots the sixth power dependence of transmittance on diameter for a circularaperature. b Scanning electron micrograph of the silicon lens and tip and the pyrex mount. c Transmission data collected by anInSb CCD array through a patterned two-bridge structure with and without the SIL.

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3.1.2 Single Crystal Films. The in-plane conductivities ofsingle-crystal semiconducting monolayers are important for

Downloaded Fsilicon-on-insulator ~SOI! @114# and 3D circuits @115#, as well asmicromachined sensors and actuators involving crystalline siliconmembranes @116#. Conduction is dominated by phonon transport,even for heavily-doped samples. Figure 13~a! shows that fornearly single-crystal silicon, phonon-interface scattering stronglyreduces the effective thermal conductivity at low temperatures.The data can be predicted using the model of Holland @117# forconduction in silicon with modifications to account for scatteringon film interfaces based on the phonon Boltzmann equation @118#.Crystalline monolayers are often bounded by amorphous films,which act as diffuse phonon emitter-absorbers owing to strongphonon scattering. When the mean free paths in neighboring filmsare comparable, which is the case in many superlattices, transportin the layers is coupled and the fraction of phonons specularlyreflected must be determined from the interfacial roughness andthe phonon wavelength. For single-crystal silicon layers doped atconcentrations higher than 1.031017 cm23 the conductivity is re-duced by scattering on impurities and free electrons @119#. Figure13~a! shows that this is particularly important at low temperatures.Measurements at 20 K on layers of thickness 3 mm with phos-phorus and boron concentrations of 1.031018 cm23 yielded re-ductions by approximately two and four, respectively, with thedifference resulting from the disparity in mass of the two impuritytypes.

The anisotropy and nonlinearity of acoustic phonon dispersionrelationships complicate predictions above 100 K for GaAs, Ge,and AlAs films, and above 200 K for silicon films owing to itshigher Debye temperature. Figure 13~b! plots room-temperaturein-plane thermal-conductivity data for silicon films in SOI sub-strates as thin as 74 nm @106#. The thermal conductivity is smallerby as much as 50 percent than the bulk value and decreases slowlywith decreasing film thickness. A graybody approximation over-predicts the data because it neglects the spectral dependence ofphonon scattering. Additional modeling lended support to the hy-pothesis that longitudinal phonons dominate transport. This hy-pothesis is consistent with the highly dispersive nature of high-frequency transverse acoustic phonons in silicon, which reducestheir group velocities.

Epitaxially-grown high-temperature superconducting films,such as YBa2Cu3O7 , EuBa2Cu3O7 , and BiSr2Ca1Cu2O8 , exhibithighly anisotropic thermal conduction properties due to theirorthorhombic unit cell and the large density of oriented imperfec-tions @120#. These films are promising for low-loss interconnectsand Josephson junctions in hybrid superconductor/semiconductorcircuits, as well as for thermal radiation detectors. Models havebeen developed for the conductivity reduction in films with the caxis oriented normal to the substrate @121,122#. These calculationsconsidered the simultaneous contribution of electrons andphonons, because both carriers are significant at temperaturesabove a few tens of Kelvin.

The reduction is substantial for films thinner than about 100 nmand is dominated by the impact of interfaces on the phonon con-tribution. The electron thermal conductivity and associated thin-film size effect are strongly reduced at low temperatures due to theincreasing concentration of Cooper pairs, which do not contributeto heat conduction.

3.1.3 Interfaces. In many cases, heat transport on shortlength-scales and time-scales is sensitive to the properties of in-terfaces. Thus, an ability to control and exploit interface thermalproperties could enable improved thermal management ofmultilayer device structures @123#, simplify the fabrication of ther-mal sensors, and increase the efficiency of thermoelectric energyconversion @124,125#. Furthermore, because thermal phonons nearroom temperature are sensitive to atomic-level structure, greaterscientific understanding of heat transport at interfaces may lead toconvenient, nondestructive methods for evaluating the microstruc-ture of interfaces: interface microstructure is a critical issue in

Journal of Heat Transferrom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Tematerials science but the high resolution electron microscopy anddiffraction needed to characterize interface structure is often dif-ficult, time-consuming and expensive.

Fig. 13 a Data for bulk silicon compared with data for siliconsingle-crystal films and polysilicon. b Predictions and data forsingle-crystal silicon films of thickness down to 74 nm at roomtemperature as a function of film thickness. c Contributionsto the thermal resistivity, 1k, as a function of temperature for adoped polysilicon layer.

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mission coefficient given by the relative density of vibrationalstates on the two sides of the interface @94,52,127#. Furthermore,

Downloaded F3.1.3a Interface thermal conductance. We first consider thethermal conductance of an isolated interface, i.e., an interface thatis separated from other interfaces by a distance that is large com-pared to the mean-free-path of the lattice vibrations that dominateheat transport in the material. In this limit, we can ignore coherentsuperposition of lattice waves reflected or transmitted by adjacentinterfaces. The conductance of isolated interfaces has a long his-tory in condensed matter physics beginning with the problem ofthe Kapitza conductance between a solid and liquid heliumanissue of great practical importance in the design of cryogenic re-frigerators for low temperature physics. Swartz and Pohl @94# givean exhaustive review of thermal boundary resistance, the inverseof interface thermal conductance, through 1988.

Some recent data for interface thermal conductance are shownin Fig. 14. Surprisingly, near room temperature, the highest ther-mal conductance for metal/dielectric interfaces measured by pico-second thermoreflectance @52#, Ti/Al2O3 and Al/Al2O3 , is only afactor of ;5 larger than the lowest conductance, Pb/diamond.Other data for metal/dielectric interfaces @94,55,126,49#, for themost part, fall between these two extremes. ~Initial experimentson epitaxial TiN/MgO, however, indicate an interface conductancea factor of ;3 larger than Al/Al2O3 , see Figs. 10 and 14.!

The diffuse mismatch model @94# provides a straightforwardstarting point for theoretical understanding of heat transport at aninterface. The key assumption of this model is that phonons arerandomly and elastically scattered at the interfaces with a trans-

Fig. 14 Comparison of selected data for interface thermal con-ductance: i individual interfaces measured by picosecondthermoreflectance 52, AlAl2O3 open diamonds and Pbc-Cfilled triangles; ii conductance of the a-GeSbTe2.5 ZnS inter-face open triangles from a multilayer sample 133, iii seriesconductance of the top and bottom interfaces of metal-SiO2-silicon structures MOS, filled circles 103. The solid lineis the calculated diffuse mismatch conductance of AlAl2O3 us-ing the Debye model; the dashed line is the theoretical predic-tion for AlAl2O3 using a lattice-dynamical calculation of amodel fcc interface 52. The right axis gives the thickness of afilm with L1 W m1 K1 that has the thermal conductancecorresponding to the left axis.

234 Vol. 124, APRIL 2002rom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Teif the Debye model is used for the density of states and modedispersion, then the thermal conductance is easily calculated fromthe velocity of sounds and atomic densities of the two materials@94,127#. An example calculation using the diffuse mismatchmodel is shown in Fig. 14; this calculation for Al/Al2O3 overes-timates the experimental data near room temperature by a factorof 5; if dispersion of the high energy vibrational modes is takeninto account, the agreement improves @52#.

The opposing limit is the assumption of no scattering at theinterface; in the long-wavelength or continuum limit, phonon re-flection and transmission is calculated from the mass density andanisotropic elastic constants of the materials. For shorter wave-lengths, microscopic calculations of the lattice dynamics areneeded; these models are typically based on idealized models ofthe interface structure and bonding @128,129#. The results oflattice-dynamical calculations @52# for Al/Al2O3 , see Fig. 14, arein reasonably good agreement with the data.

While the data in Fig. 14 represent a significant advance, manymore systematic studies will be needed to develop a comprehen-sive understanding; for example, experiments have not yet ob-served either the high or low extremes of conductance predictedby theory @52# and have not yet isolated contributions to heattransport at interfaces by anharmonicity, electron-phonon cou-pling @130#, interface disorder @131# or altered interface bonding@132#.

3.1.3b Multilayer thin films. Figure 14 also contains data forinterface conductance extracted from studies of multilayers @133#and thin film samples @49,126# measured using the 3v method. Ifthe thermal conductivity of a thin layer of disordered material isindependent of layer thickness d, i.e., if phonons with mean-free-paths comparable to d make a negligible contribution to heattransport, and that the microstructure of the thin layer is approxi-mately independent of d, the measured conductance of amultilayer sample is the parallel sum of the thermal conductanceof the individual layers and the thermal conductance of the inter-faces @49,134,133#. Data for GeSbTe2.5 /ZnS interfaces ~a compo-nent of phase-change data-storage media! nearly overlap with datafor the series conductance of the metal/SiO2 and SiO2 /Si inter-faces of metal-oxide-semiconductor structures. These data formetallized SiO2 layers on Si are in good agreement with previousroom temperature measurements using nanosecond thermoreflec-tance @48#.

While GeSbTe2.5 /ZnS multilayers show relatively small ther-mal conductance, our studies of multilayers of disordered oxideshave failed to reveal a significant effect of interfaces: the thermalconductivity of ZrO2 :Y2O3 /SiO2 @135# and ZrO2 /Y2O3 @126#multilayers are almost independent of layer thickness with layerthickness as small as 4 nm. Apparently, the lattice vibrations ofthese oxides are sufficiently alike that the thermal conductance istoo large to be measured by this approach. Multilayer coatings@136# do not, therefore, appear to be a promising route for lower-ing the conductivity of thermal barriers. Significant decreases inconductivity have been observed recently in nanocrystallineZrO2 :Y2O3 @137# and attributed to the finite thermal conductanceof grain boundaries.

3.1.3c Epitaxial superlattices. For a short-period superlat-tice with coherent interfaces, we must reevaluate our assumptionsfor isolated interfaces and consider superposition of lattice wavestransmitted and reflected by the interfaces. Thermal transport insuperlattices with perfect interfaces has been discussed in a seriesof papers using increasingly refined models of the lattice dynam-ics @138,139,140,141#. The general conclusion of this work is thata superlattice structure strongly reduces the average group veloc-ity of phonons in the through-thickness direction andbecause

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insufficient to explain the data. Approaches that combines bothmechanisms, wave-interference and diffuse scattering @143#, per-

Downloaded Fthe thermal conductivity scales as the square of the groupvelocitydramatically suppresses the thermal conductivity incomparison to the bulk components.

This suppression of thermal conductivity produced by zone-folding and phonon confinement cannot, however, fully explainthe data, see Fig. 15. The observed reduction in conductivity is toolarge to be easily explained by the theories of perfect interfacesand both Si-Ge @142# and GaAs @54# superlattices show substan-tial decreases in conductivity with decreasing superlattice period;theory predicts that the conductivity should increase slowly withdecreasing period @143#.

Theory @93# based on incoherent ~particle-like! scattering ofphonons at interfaces predicts a superlattice thermal conductivitythat decreases with superlattice period. Particle-based theory @93#can more easily explore effects of disorder but this approach re-quires approximations and assumptions concerning phonon dis-persion, elastic vs. inelastic scattering, and specular vs. diffusescattering that may be difficult to justify microscopically.

Interface disorder introduces diffuse scattering but disorder in asuperlattice is typically modest and limited to interface roughness~a finite density of interface steps! and substitutional alloying atthe interfaces caused by surface segregation during growth. Un-fortunately, this physical and chemical interface roughness in anepitaxial superlattice is sensitive to the material, growth method,growth temperature, and deposition rateand is notoriously dif-ficult to characterize. Furthermore, asymmetry of the interface isoften pronounced, i.e., the growth of component A on the surfaceof component B creates a different interface than the growth of Bon A. Strained-layer superlattices such as Si-Ge can harbor highdensities of crystalline defects when the layer thickness exceedsthe critical thickness for the extension of misfit dislocations @104#;growth on relaxed buffer layers @104# reduces the density ofthreading dislocations but the misfit density will still be largewhen the critical thickness is exceeded.

Since the through-thickness conductivity of a short-period su-perlattice is smaller than the corresponding alloy @142,170#, seeFig. 15, substitutional interface disorder acting alone is probably

Fig. 15 Selected data for the through-thickness thermal con-ductivity of superlattices with bilayer periods of 5 nm; datafor GaAs-AlAs open circles, 5.67 nm period 54 were mea-sured by picosecond thermoreflectance; data for Si-Ge opentriangles, 5 nm period 142; filled circles, 4.4 nm period 104and InAs-AlSb open diamonds, 6.5 nm period 170 were mea-sured using the 3v method. Data for Si0.85Ge0.15 142 andGa0.4Al0.6As 144 alloys are included for comparison.

Journal of Heat Transferrom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Tehaps acting independently on different parts of the phonon spec-trum, may be a more successful route to a microscopic theory. Wenote that emerging data on the anisotropy of superlattice thermalconductivity @145#, i.e., the ratio of the in-plane to the through-thickness conductivity, may provide additional constraints on the-oretical models.

3.1.4 Polycrystalline Films. In polycrystalline films, phononscattering on grain boundaries and related defects dominates overboundary scattering. Doped polysilicon films are common inMEMS and as the gate terminal in silicon field-effect transistors.The grain structure and spatial impurity distribution dependstrongly on the temperature and duration of annealing. Polysiliconconductivities are reduced compared to those of pure crystallinefilms @146#, but the roles of impurities and grains are difficult toseparate. Recent work studied films with grain sizes between 300and 500 nm, extracted using transmission electron microscopy,and boron or phosphorus concentrations up to 4.131019 cm23,measured using secondary ion mass spectroscopy ~SIMS!@147,148#. Figures 13~a! and 13~c! show that the conductivity isreduced strongly at all temperatures compared to similarly-dopedsingle-crystal silicon layers, which illustrates the importance ofgrain boundary scattering. The data were predicted by reducingthe mean free path using Matthiessens rule @149# and a grain-boundary free path given by B/dG , where dG is the grain size andB is governed by the grain shape and grain boundary reflectioncoefficient. The data at 300 K are well approximated by

k~dG ,n !513 CvS A1dG 1A2niD

21

,

where the phonon velocity and heat capacity are v56170 m s21and C51.653106 J m23 K21, the grain size dG has units of nm,and the impurity concentration ni has units of cm23. The constantA1 is 2.8931010 m2 and A2 is 3.20310213 or 21.12310214 m2 for boron and phosphorus, respectively.

For polycrystalline diamond films the orientation and minimumsize of grains are governed by the details of the deposition pro-cess, in particular the nucleation technique, the substrate tempera-ture, and the composition of the process gases. Verhoeven et al.@150# observed a particularly large degree of anisotropy for mi-crowave CVD diamond films with predominantly heteroepitaxialgrains, with the in-plane conductivity smaller by about one orderof magnitude than the out-of-plane conductivity as shown in Fig.16. The free path B/dG is overly simplistic for diamond film dataowing to the large concentration of other imperfections. Becauseimperfections populate primarily at grain boundaries, the phononscattering rate can still be coupled to the grain size dG . Thisconcept was quantified using a dimensionless grain-boundaryscattering strength @151#, which is governed by the scatteringcross sections and number density per unit grain boundary area ofimperfections. For this model, the scattering rate is dictated byboth the size of grains and the density of imperfections per unitgrain boundary area.

3.1.5 Amorphous Oxide and Organic Films. For highly-disordered films, such as amorphous glasses and organic materi-als, the process-dependent material structure and stoichiometryinfluence the thermal conductivities and volumetric heat capaci-ties. These films are common as passivation and thermal insula-tion in MEMS and integrated circuits. Silicon dioxide and siliconnitride films are often deposited at low temperatures to avoid dif-fusion or other failure mechanisms in micromachined structures.Lower deposition temperatures for silicon dioxide are known toreduce the mass density, which decreases the thermal conductivity@152# but increases the volumetric heat capacity owing to the largeconcentration of silanol @107#. Following a high-temperature an-neal, the root-mean-square deviations of bond lengths, which can

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study thermal transport in these nanowires as well, which can bevery different from those of bulk materials. For example, Schwabet al. @163# obsercated one-dimensnon conductanceconstant, T is theregime, a one-dia phonon wavegtransport in nanoductors mainly bcantly modified dthe presence ofThese result in mtwo transverse abulk semiconducmodify the groupThe phonon lifesources. First, thcause selectionvector relations

d heater struc-of nanowires.

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236 Vol. 124 of the ASME

Downloaded From: http://heattraved quantum thermal conductance in nanofabri-ional nanostructures, where the quantum of pho-is go5p2kB

2 T/(3h), where kB is the Boltzmanntemperature, and h is the Planck constant. In this

mensional nanostructure behaves essentially likeuide similar to optical ones for light. Phononwire can be different from that in bulk semicon-ecause the dispersion relation could be signifi-ue to confinement in two directions. In addition,a surface can introduce surface phonon modes.any different phonon polarizations other than thend one longitudinal acoustic branches found intors. Such changes in the dispersion relation canvelocity and the density of states of each branch.time also changes and this arises from twoe phonon-phonon interactions can change be-

rules based on energy conservation and wave-depend on the dispersion relation. Second,

Fig. 17 Electron micrograph of the suspendeture used for measuring thermal conductivityAlso laid across the heater is a multiwall cabundle 150 nm in diameter. The Pt heater coilsuspended islands is heated and its temperatuby resistance thermometry. The temperature ofincreases due to conduction through the nanoalso measured by resistance thermometry. Thdifference for a known heat flow rate from thecan be used to estimate the thermal conductivwire.

, APRIL 2002 Transactionsserve as a measure of the degree of disorder, diminish consider-ably and approach those of thermally-grown silicon-dioxide films.The thermal properties of organic films are expected to be highlysensitive to their chemical composition and structural configura-tion. Figure 16 shows that polyimide films exhibit anisotropicconductivities due to the partial alignment of molecular strands inthe film plane during spin coating. The modeling of heat conduc-tion in disordered films is far more approximate than that of crys-talline films, and is limited to estimates of the minimum conduc-tivity @153# and simple models for the anisotropy of fully-amorphous stranded organic films @97#.

3.2 Nanowires. Much of the past effort on synthesizinglow-dimensional structures has focused on 0-D ~quantum dots!@154# and two-dimensional ~quantum wells and heterostructures!nanostructures. Except for carbon nanotubes @37#, one-dimensional structures have received relatively little attention.This is mainly due to challenges in synthesis of single-crystalnanowires, which are important for determining structure-propertyrelations. Recently, however, there have been some major ad-vances in growth of monocrystalline semiconductor nanowires@155,156,157,158# and integrating them into devices@159,160,161#. While most of the current research is being fo-cused on electronic and optical properties of nanowires @162#, themanipulation of their thermoelectric properties through quantumconfinement of electrons is extremely promising for developingsolid-state energy conversion devices @46#. In this case, one must

Fig. 16 The sketches indicate the impact of gransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Teboundary scattering can be much stronger in nanowires ~550 nmdiameter! than in bulk semiconductors @164#. While there havebeen some theoretical studies on phonon conduction in nanowires@165,166#, experiments have lagged behind. Recently, there havebeen some attempts @167# to measure the thermal conductivity ofa collection or a mat of carbon nanotubes. However, due to weakcoupling between carbon nanotubes, the actual thermal conductiv-ity was found to be much lower than that predicted @168#. Toaccurately measure thermal properties of nanowires and comparethem with theoretical studies, one must develop techniques tostudy the thermal behavior of isolated nanowires.

Figure 17 shows a micromachined structure that was developedby Shi @33,169#. The device is a suspended structure consisting oftwo adjacent silicon nitride (SiNx) membranes or islands sus-pended with three 200 mm long and 2 mm wide silicon nitridebeams. One 30 nm thick, 200 nm wide, and 150 mm long plati-

in and molecular orientation on the anisotropyrms of Use: http://www.asme.org/about-asme/terms-of-use

perature. The underlying principles, recent advances, and futurechallenges and research directions of scanning thermal micros-

Downloaded Fnum ~Pt! heater/thermometer coil is built on each island. A nano-wire can be placed in between and bridge the two suspendedislands by several methods. A general method is to deposit a dropof aqueous solution containing nanotubes and nanowires on a wa-fer with densely packed suspended structures. This method hasbeen successfully used to place SWCNs or MWCNs on the sus-pended structures. In the case of SWCN, nanotubes can also begrown between two islands using chemical vapor deposition~CVD!. For this method, catalyst particles consisting of Fe,Al2O3 , and Mo are first deposited on the islands from aqueoussolution. Then the device is placed in a 750900C CVD chamberwith methane gas flow. There is a good chance that one or severalSWCNs can be grown and bridging the two islands.

To measure the thermal conductivity, one of the islands isheated to a temperature, Th , which is determined by measuringthe resistance of the Pt wire on the island. Conduction through thenanowire heats the sensing island to a temperature Ts , which isalso measured by resistance thermometry. By noting the powerdissipated by the heater, Qh , and estimating the thermal conduc-tance of the suspended legs connecting the islands in the absenceof a nanowire, the conductance of the nanowire Gn can be calcu-lated in a vacuum environment. Figure 18 shows the plot of ther-mal conductivity of a multiwall carbon nanotube as a function oftemperature measured by this technique. The T2 temperature de-pendence suggests that MWCN thermally behaves like a two-dimensional solid and has a thermal conductivity of about 3000W/m-K at room temperature. This is approximately half of what istheoretically predicted @168#. Note that the measured value in-cludes the contact resistance between the nanotube and the twoislands, although it is predicted to be lower than the resistance ofthe tube itself @169#. Estimation of the contact resistance has re-mained a challenge. Nevertheless, it is envisioned that this tech-nique can be used to measure thermal conductivities of a varietyof semiconducting or metallic nanowires that are of technologicalimportance.

4 Closing RemarksThis paper provides a critical review of two interrelated areas of

research, namely, microscale thermometry and thermal transportin solid micro/nanostructures. The technology for measuring tem-perature and studying thermal transport is sufficiently advancedthat one can now approach scales on the order of the mean freepath and relaxation times of electrons and phonons at room tem-

Fig. 18 Temperature dependence of thermal conductivity of amultiwall carbon nanotube. The dashed line corresponds to aT2 temperature dependence. The thermal conductivity is about3000 Wm-K at room temperature, which includes the contactresistances at the two islands.

Journal of Heat Transferrom: http://heattransfer.asmedigitalcollection.asme.org/ on 10/04/2015 Tecopy, picosecond thermoreflectance, and scanning optical ther-mometry are discussed here. Experiments based on these tech-niques have led to fundamental understanding of electron, phononand photon interactions and transport in both bulk and micro/nanostructured solids as well as solid-state devices. While thesetechniques are being developed to study thermal phenomena atmicroscales, it is clear that a fundamental understanding of mi-croscale heat transfer is necessary to make further advances inthese techniques. Much work remains to be done for further re-fining and combining these techniques. For example although onecan now study short length and time scales thermal phenomenaindependently, it is currently very difficult to study thermal phe-nomena at small length and time scales simultaneously. In addi-tion, although significant advances have occurred over the pastdecade, these techniques are limited to only a few laboratories.There needs to be more effort in making these techniques suffi-ciently robust and accessible so that the barrier for researchers toenter this field is greatly reduced.

This paper presents a critical review of thermal transport intwo-dimensional and one-dimensional solid microstructures. Inparticular, phonon transport in single crystal, polycrystalline, andamorphous oxide and organic thin films is elucidated. The role ofinterfaces in multilayered polycrystalline and epitaxial thin filmsis also discussed. Finally, thermal transport in carbon nanotubesis presented and the opportunities for further research on nano-wires are explored. For single crystal films, the effect of filmthickness on thermal conductivity in materials such as Si is nowfairly well understood and the thermal conductivities can be pre-dicted using Boltzmann transport theory. For other materials,however, the non-linearity of the acoustic phonon dispersion rela-tions makes it difficult to predict thermal conductivities at roomtemperature. In fact, in many materials the contribution of eachphonon branch, longitudinal and transverse, to the thermal con-ductivity is largely unknown. For polycrystalline Si and diamondfilms, grain boundary scattering has been found to strongly reducethermal conductivity. The value depends on the grain structure,which in turn depends on the growth process. Thermal propertiesof organic films are very sensitive to the chemical compositionand structural configuration. For example, spin coating a polyim-ide film partially aligns the polymer chains along the film plane,which produces anisotropic thermal conductivity. Simple modelshave been developed to predict their thermal conductivity al-though detailed models based on fundamental understanding ofpolymer structure and chemistry remains to be developed.

Thermal transport across solid-solid interfaces remains one ofthe most challenging problems. While acoustic and diffuse mis-match models have been developed, their predictions rarely agreewith experimental data. These models are phenomenological anddo not isolate contributions to heat transport by anharmonicity,electron-phonon coupling, interfacial disorder and chemistry. Ex-periments have shown that interfaces play a very important role inepitaxial superlattices, where interface roughness and alloying in-troduces phonon scattering and thereby reduces the effective ther-mal conductivity. In some cases, it has been found that the thermalconductivity reduction exceeds even that offered by a collection ofindividual interfaces. It is conjectured that the wave interferencefrom multiple interfaces, which produces phonon bandgaps andthereby reduction of group velocity, may play an important role.However, what is clear is that fundamental understanding of theindividual and collective role of interfaces on thermal conductiv-ity is largely missing.

While most of the paper discusses thermal transport in two-dimensional confined structures ~films!, it briefly discusses pho-non transport in one-dimensional nanostructures, such as carbonnanotubes. It is expected that these and other nanowires are likely

APRIL 2002, Vol. 124 237rms of Use: http://www.asme.org/about-asme/terms-of-use

to play an important role in nanostructured materials and devices,and therefore ought to be thermally studied. Confinement in an

Access Communication Systems, IEEE J. Sel. Areas Commun., 8, pp. 945947.

Downloaded Fadditional dimension might offer avenues of pushing the thermalconductivity to the extremes, as exemplified in carbon nanotubes.However, theoretical and experimental studies in this area haveonly just begun, and much remains to be understood.

AcknowledgmentsDGC was supported by NSF Grant No. CTS 99-78822 and the

U.S. Department of Energy, Division of Materials Sciences underAward No. DEFG02-ER9645439, through the Frederick Seitz Ma-terials Research Laboratory at the University of Illinois at Urbana-Champaign. Picosecond thermoreflectance measurements and filmthickness characterization used the Laser Facility and the Centerfor Microanalysis of Materials, University of Illinois. DGC thanksK. E. OHara ~Integrated Micromachines! for invaluable assis-tance with the picosecond experiments and H. J. Maris, G. Chen,P. M. Norris, and G. D. Mahan for helpful discussions.

KEG appreciates the help of D. A. Fletcher and A. D. McCon-nell of Stanford University in preparing the sections on the SILand polysilicon films, respectively. KEG also appreciates themany technical contributions of former students including M.Asheghi of Carnegie Mellon University, K. Kurabayashi of TheUniversity of Michigan, Y. S. Ju of IBM Corporation, M. N. Tou-zelbaev of AMD, and P. G. Sverdrup of Intel Corporation. Spon-sorship was provided from the SRC through tasks 751 and 754.

AM greatly appreciates the support of the NSF, the DOE ~En-gineering Division, Basic Energy Sciences!, and DARPA for theirsupport. AM acknowledges the many contributions of severalformer students and post-docs, in particular, Jie Lai ~MicrosemiCorp.!, Ke Luo ~TriQuint Semiconductor!, Madhavi Chandrac-hood ~Applied Materials!, Osamu Nakabeppu ~Keio Univ.!, JohnVaresi ~Raytheon!, and Li Shi ~IBM Research Division!. He alsothanks Paul McEuen, Sergie Plyasunov, Adrian Bachtold andPhilip Kim ~Physics, UCB! for their help with nanotube relatedresearch.

NomenclatureD 5 thermal diffusivityf 5 modulation frequency of the probe beam

G 5 interface thermal conductance ~per unit area!k 5 imaginary part of the complex index of refractionl 5 thermal diffusion length

n 5 complex index of refractionq 5 summation index

DR1(t) 5 change in optical reflectivity at time t following asingle pump pulse

t 5 timeT 5 temperature

DT1(t) 5 temperature excursion at time t following a singlepump pulse

DT(n) 5 frequency domain temperature responseVLI 5 lock-in signal expressed as a complex numberV in 5 in-phase lock-in signal

Vout 5 out-of-phase lock-in signall 5 laser wavelength ~800 nm for Ti:sapphire!n 5 frequencyt 5 time between mode-locked optical pulses

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