3FWJFX SUJDMF …

Hindawi Publishing CorporationISRNMechanical EngineeringVolume 2013 Article ID 682586 19 pageshttpdxdoiorg1011552013682586

Review ArticleThermal Transport across Solid Interfaces with NanoscaleImperfections Effects of Roughness Disorder Dislocations andBonding on Thermal Boundary Conductance

Patrick E Hopkins

Department of Mechanical and Aerospace Engineering University of Virginia Charlottesville VA 22904 USA

Correspondence should be addressed to Patrick E Hopkins phopkinsvirginiaedu

Received 20 November 2012 Accepted 7 December 2012

Academic Editors Y He T Ohara and Z Yu

Copyright copy 2013 Patrick E Hopkins is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e efficiency in modern technologies and green energy solutions has boiled down to a thermal engineering problem on thenanoscale Due to themagnitudes of the thermalmean free paths approaching or overpassing typical length scales in nanomaterials(ie materials with length scales less than one micrometer) the thermal transport across interfaces can dictate the overall thermalresistance in nanosystems However the fundamental mechanisms driving these electron and phonon interactions at nanoscaleinterfaces are difficult to predict and control since the thermal boundary conductance across interfaces is intimately related tothe characteristics of the interface (structure bonding geometry etc) in addition to the fundamental atomistic properties of thematerials comprising the interface itself In this paper I review the recent experimental progress in understanding the interplaybetween interfacial properties on the atomic scale and thermal transport across solid interfaces I focus this discussion specicallyon the role of interfacial nanoscale ldquoimperfectionsrdquo such as surface roughness compositional disorder atomic dislocations orinterfacial bonding Each type of interfacial imperfection leads to different scattering mechanisms that can be used to control thethermal boundary conductance is offers a unique avenue for controlling scattering and thermal transport in nanotechnology

1 Introduction

As the worldrsquos population and power demands increase ourtechnological solutions continue to rely on power-hungryapplications e shear population increase in the worldalong with skyrocketing electricity power and transporta-tion needs in emerging markets has led to necessary mini-mum levels of energy for sustainable growth and livelihood ofhuman kinde trends in energy use aremarked by our soci-etyrsquos continued advancement of technology communicationtransportation and quality of life For example in 2010 theUnited States used an estimated 287 trillion kW-h of energy[1] ese massive energy demands both in the USA andabroad are a necessary requirement for continued capabilitiesand qualities of life that have been made possible withthe advancement of technologies For example increasedcomputing needs (internet cloud computing and wirelesscommunications) although more energy efficient in therecent past have led to an increase in data center energy usage

by roughly a factor of six over the past decade [2] althoughtechnologies have become more energy efficient due to theshear increase in the numbers of users and informationenergy consumption and needs are skyrocketing leading tothese astounding levels of energy usage

e primary fuel for this electricity generation comesfrom coal a nonrenewable resource that leads to 2250 lbsMW-h of carbon dioxide 13 lbsMW-h of sulfer dioxide and6 lbsMW-h of nitrogen oxides [2] is leads to 126 billionlbs of CO2 emissions along with 73 and 34 million lbs ofsulfer dioxide and nitrogen oxides respectively In additionmore thanhalf (57) of the used energy in theUSA is rejectedas heat where only 43 of the consumed energy is actuallyused (work) ese stiing statistics create two major foci forenergy research (1) improving the efficiency of the energyusage to reduce the net wasted heat and (2) reducing theharmful emissions of the technologies generating the energyImproving energy efficiency of devices has been an active areaof material research for the last few decades stemming from

2 ISRNMechanical Engineering

the drive of the computing industry to maintain Moorersquos Law[3] and recently focused on mitigating the thermal transportin the nanoscale devices [4 5] Recent novel approachesin reducing the net wasted heat have focused on wasteheat recycling via thermoelectric power generation [6ndash8]ermoelectric solutions also show promise in reducingharmful emissions in the grid as the entirely solid statemakeup of thermoelectrics requires no moving parts cyclesor input fuel other than heat ese thermoelectric solutionscombined with ldquorenewable sourcerdquo technologies (technolo-gies that do not rely on coal as a fuelmdashsuch as photovoltaics[9 10]) could substantially reduce the carbon footprint of theenergy grid

e issue with these technologies lies in their operationefficiencies which at this stage of research have boileddown to a thermal engineering problem on the nanoscaleFor example power dissipation limits the performances ofelectronic systems from individual microprocessors to datacenters In silicon-based microprocessors (by far the mostcommonly implemented technology) operation beyond afew GHz is not possible due to on-chip power densitiesexceeding 100Wcmminus2 higher than a typical hotplate andfar greater than typical cooling capabilities [4] Even greaterpower densities have been observed in RF and amplierdevices using IIIndashV or group IV compound semiconductors(eg GaAs or SiC) [11] ese huge power densities leadto hot-spots and device failure warranting the need forincreased thermal conduction to mitigate the heat load Asanother example current thermoelectric solutions for wasteheat recovery are far outperformed by traditional solutionssuch as heat exchangers with heat pumps or heat enginecycles such as the Organic Rankine or Kalina cycles [6] mak-ing major implementations of the more-carbon-footprint-friendly thermoelectric devices impractical Improvement inthe thermoelectric efficiency can be achieved by controlledreduction in thermal transport of the thermoelectric mate-rials since the thermoelectric efficiency (quantied by theFigure of Merit ZT) is inversely related to the thermalconductivity Recentmassive improvements in ZT in thermo-electric materials have been discovered by nanostructuring[8 12ndash15] leaving the research problem to improve energyefficiency centered around understanding thermal transporton the nanoscale As both increases and decreases in thermalconductivity are required for different applications the abilityto control and tune the thermal properties of materials istherefore the ultimate goal to improve device efficiency andimprove energy consumption trends

is desired thermal control in nanosystems howeveris not a trivial task Due to the magnitudes of the thermalmean free paths approaching or overpassing typical lengthscales in nanomaterials (ie materials with length scales lessthan 1 micron) the thermal transport across interfaces candictate the overall resistance in nanosystems is leads to aregime in which unique yet relatively unknown mechanismssuch as interfacial scattering ballistic transport and wave-effects offer potential new degrees of freedom in the ther-mal engineering of material systems [5 16] However thefundamental mechanisms driving these electron and phonon

Perfect

interface

Substrate

Film

Imperfect

interface

Film

Substrate

F 1 Schematic of a perfect and an atomically imperfectinterface A perfect interface is an atomically abrupt coherentinterface between two solids in intimate contact e atomicallyimperfect interface depicted here occurs when the interface isnot well dened due to atomic disorder (both structural andcompositional) resulting in roughness and impurities around theinterface

interactions at nanoscale interfaces are difficult to predictand control since the thermal boundary conductance acrossinterfaces (oen called the Kapitza conductance) ℎK [17 18]is intimately related to the characteristics of the interface(structure bonding geometry etc) in addition to the funda-mental atomistic properties of the materials comprising theinterface itself is interplay between interfacial propertiesand thermal boundary conductance can lead to additionalcarrier scattering mechanisms and phenomena that will notonly change the thermal processes but can also be used tosystematically manipulate and control thermal transport innanosystems

As an example consider a ldquoperfectrdquo interface and anatomically disordered ldquoimperfectrdquo interface depicted inFigure 1e imperfect interface introduces roughness alongwith atomic imperfections that cause additional thermal scat-tering which can change ℎK as I have discussed in previousworks [19ndash24] and will detail throughout this paper eproperties of this interface dictate the additional scatteringmechanisms erefore an understanding of the effects ofinterfacial imperfections and other ldquonon-idealitiesrdquo (suchas roughness disorder and bonding) on thermal transportacross complex nanoscale interfaces is warranted to fullydevelop the potential to control electron and phonon thermalprocesses in nanosystems

To this end in this paper I will review recent advance-ments in the underlying physics of thermal boundary con-ductance at solid interfaces focusing specically on therole of interfacial nanoscale ldquoimperfectionsrdquo such as surfaceroughness compositional disorder atomic dislocations orinterfacial bonding In the next section I present an overviewof thermal boundary conductance anddiscuss pertinentmea-surements of thermal boundary conductance between twosolids to put this thermophysical quantity into perspectiveFollowing this background I discuss time domain thermore-ectance in Section 3 TDTRhas become a common effectiveexperimental technique to accurately measure the thermalboundary conductance between two solids In Section 4

ISRNMechanical Engineering 3

I will discuss the mathematical semiclassical formalisms forpredicting thermal boundary conductance which are simpleanalytical expressions that can be used to understand theelectron or phonon transmission across interfaces Followingthis the remainder of the sections will focus on examiningthe effects of various interfacial imperfections on either theelectron- or phonon-dominated thermal boundary conduc-tance across interfacesese sectionswill focus on the effectsof interfacial interspecies mixing on electron and phononthermal boundary conductance (Sections 5 and 6 resp)and roughness dislocations and bonding effects on phononthermal boundary conductance (Sections 7 8 and 9 resp)

2 Background ThermalBoundary Conductance

e thermal transport across solid interfaces is quantied bythe thermal boundary conductance ℎK or Kapitza conduc-tance [18] which quanties the ecacy of heat ow acrossan interface Mathematically the thermal boundary conduc-tance is the proportionality constant that relates the heat uxacross an interface to the temperature drop associated withthat interfacial region via

ℎK =119902119902intΔ119879119879

(1)

where 119902119902int is the thermal ux across the solid interfacialregion and Δ119879119879 is the temperature drop across the interfacialregion It is also oen intuitive to conceptualize the inverse ofthis quantity the thermal boundary resistance 119877119877K = 1ℎKwhich can be paralleled to an electrical resistance networkNote that thermal boundary conductance is discussed herein terms of twomaterials in intimate contact therebymakingthis inherently different than a ldquocontact conductancerdquo whichis related to the voids or lack of contact between two solids

Measurements of the thermal boundary conductancebetween two solids typical occur in a thin lm geometry or atultralow temperatures In both of these cases the resistancedue to the interface is a major resistance in the measuredsystem of interest thereby allowing for accurate measure-ment sensitivity One of the rst comprehensive reviewsof thermal boundary conductance was detailed by Swartzand Pohl in 1989 [17] and their review mainly focused onlow temperature measurements and phenomena Recentlywith the advances in sensitive measurement techniquesfor interrogating thermal transport on the nanoscale [25]accurate measurements of thermal boundary conductancein nanosystems and at elevated temperatures have beenachieved [4 5]

As the goal of this paper is to discuss the effects of inter-facial ldquonon-idealitiesrdquo on thermal boundary conductanceonly a subset of the measurements of thermal transportacross various interfaces will be discussed here e readeris referred to [4 5 26] for overviews of thermal boundaryconductance at ldquosmoothrdquo or seemingly ldquoperfectrdquo interfacesHowever it is informative to discuss the limits of thermal

75 100 200 300 400 500

Temperature (K)

1

10

100

1000

10000= 01 nm

PdIr

AlCu

TiNMgO AlSi(roughness)

AlSLG

(bonding)= 10 nm

GaSbGaAs(dislocations)

Bidiamond

= 100 nm = 1000 nmTh

erm

al b

ou

nd

ary

con

du

ctan

ce

(MW

mminus

2K

minus1) = 1 nm

F 2ermal boundary conductance at various solid interfaces(PdIr [27] AlCu [28] TiNMgO [29] AlSi [19 21] Alsinglelayer graphene (SLG) [24] GaSbGaAs [23] and Bidiamond [30])Typical values span over three orders ofmagnitude For comparisonthe equivalent conductances of various thicknesses of SiO2 areshown as the solid lines

boundary conductance at ldquoperfectrdquo interfaces and the dif-ference between electron-dominated andphonon-dominatedthermal boundary conductances

Figure 2 shows the thermal boundary conductance asa function of temperature for various interfaces in whichℎK is dominated by either electron or phonon scattering[21 23 24 27ndash30] e TiNMgO and Bidiamond datarepresent the highest and lowest phonon-dominated thermalboundary conductance ever measured respectively [29 30]Note that the electron-dominated thermal boundary conduc-tances (AlCu and PdIr) are roughly one order of magni-tude higher than the phonon-dominated thermal boundaryconductances and have a steeper increase with temperature[27 28] Along with these data the equivalent conductancesof various thicknesses of SiO2 are indicated by the solidlines this equivalent conductance is calculated by ℎSiO2

=120581120581SiO2

119889119889 where 120581120581SiO2is the thermal conductivity of SiO2 and

119889119889 is the thickness as indicated in [31] erefore at roomtemperature the thermal boundary conductance between aPdIr interface offers the equivalent resistance as roughly 1Aringof SiO2 e highest phonon-dominated thermal boundaryconductance offers the same resistance as sim2 nm of SiO2nearly 20 times greater resistance than the electron-electron-dominated interface At the lower end of the spectrum


the phonon transport across a Bidiamond interface isequivalent to the resistance of sim200 nm of SiO2 Clearly innanomaterials and composites the thermal boundary con-ductance across solid interfaces can offer amajor impedimentto heat ow that can dominate the thermal transport in theoverall system

e additional data in Figure 2 (AlSi Alsingle layergraphene (SLG) and GaSbGaAs interfaces) represent ourrecent measurements to examine the effects of some inter-facial imperfections on thermal transport across interfaces[21 23 24] e specic interfacial imperfection that wasinvestigated in each material case is indicated by labelsin Figure 2 ese along with other measurements willbe discussed in detail throughout this paper Briey thesethree example interfaces show the effects of either roughness(AlSi) bonding (AlSLG) or dislocations (GaSbGaAs) onthe thermal boundary conductance In the AlSi case arougher substrate leads to a decrease in thermal boundaryconductance [21] At the AlSLG interfaces by increasingthe bonding between otherwise nonreacting Al lms andSLG substrates Al can bond covalently to the SLG therebyincreasing the thermal transport [24] At the GaSbGaAsinterfaces the large lattice mismatch between the GaSb andGaAs leads to strain dislocations e number of dislo-cations can be controlled via the growth technique andwe observed that making a GaSbGaAs interface with ahigher number of dislocations per unit area results in alower ℎK ese effectsmdashroughness dislocations and bond-ingmdashwill be discussed in more detail in Sections 7 8 and9 respectively However it is apparent in this preliminaryoverview that the thermal boundary conductance betweentwo solids is not only a function of the materials comprisingthe interface but has a strong dependency on the atomicproperties of the interface erefore with careful nanoscaleengineering of the interfacial properties the ability to tunethe thermal boundary conductance between two solids canbe achieved In the remainder of this paper I will discussspecic experimental results and accompanying theory onthe thermal transport across imperfect interfaces betweentwo solids including more detail on the specic imperfectinterfaces discussed above However before delving into thetransport physics driving thermal boundary conductanceacross imperfect interfaces measurements of ℎK with timedomain thermoreectance will be detailed in the followingsection

Time omain ThermoreectanceAn Effective Tool for Measuring ThermalBoundary Conductance between Two Solids

Over the past few decades contact-based techniques orexperimental platforms based on resistive thermometryapparati have made substantial progress in measuring thethermal transport in nanosystems due to advances innanofabrication and increasingly sensitive detection schemes[25] Common resistive thermometry techniques that havebeen used to determine temperature rises and heat transportproperties in nanosystems are steady-state resistance-based

techniques [32ndash37] and the 3120596120596 technique [31 38ndash43] esemethods are based on the fact that the electrical resistivityof a sample or test strip is related to the temperature changtherefore by monitoring the change in electrical resistivity ofa reference the change in temperature can be determinedWith precise knowledge of geometry of the nanosystem ofinterest the thermal conductivity can be determined from themeasured temperatures e advantages of these techniquesare that they are direct measurement of the temperature riseand can be directly related to the thermal conductivity ofthe sample of interest through only a carefully calibratedelectrical resistivity reference and precise knowledge of thesample geometry e disadvantages however in addition tothe potential uncertainty in sample geometries are that thesetechniques require intimate contact between the electricalleads and the sample platform of interest Inherent in thesecontacts are both electrical and thermal contact resistancesthatmust be accounted for [44] In addition to thesemeasure-ment difficulties direct measurement of the thermal bound-ary conductance between two materials of interest is verydifficult without the use of multiple samples For exampleif one were to use a steady-state or frequency domain (suchas 3120596120596) electrical resistivity technique to measure the thermalproperties of a bilayer sample (say thin lm on thin lmon substrate) the measured data would represent the totalthermal resistance of the sample Although the individuallm properties could be inferred from multilayer thermalmodeling [45] a ldquoheat capacity-lessrdquo thin interface could notbe differentiated from the individual layers is is not thecase with time domain techniques however

Time domain techniques for measuring the thermaltransport of materials have proven invaluable for monitoringthe thermal conductivity and thermal boundary conductanceof multilayer structures due to the fact that the thermal effu-sivity or thermal diffusivity of a material responds differentlythan the thermal transport across an interfacial region in thetime domain Several time domain techniques have provenquite useful in measurement of these interfacial thermalproperties including laser ash diffusivity [46] transientelectrothermal techniques [47] and thermoreectance tech-niques [48 49] With recent progress in pulsed laser systemsthermoreectance techniques employing short pulses havebecome a robust metrology to accurately determine the ther-mal transport in nanosystems as I outline in the followingese thermoreectance techniques which rely on the fun-damentals of thermoreectance spectroscopy of materialscan offer ultrafast characterization of the temperature changeof materials in the time domain which alleviates many of theradiative loss optical effusivity and contact issues that plagueother techniques

In thermoreectance spectroscopy an oscillating per-turbation is applied to the sample of interest and the ACcomponent of the reected light is monitored [50ndash57] Forexample in a standard pump-probe geometry the pump isa modulated laser beam or train of laser pulses that is usedto optically and thermally excite some sample at frequency119891119891 while the probe is used to interrogate the small changesin reectivity of the sample caused by the pump excitationat frequency 119891119891 To back out heat transport characteristics


using thermoreectance the majority of optical perturbationthat causes a change in reectivity must be directly due to athermal perturbation For this reason metallic systems areideal to transduce the change in reectivity into the changein temperature due to the large electronic number densityaround the Fermi level inmetalsis resulting thermoreec-tivity change in a metal is related to an electron populationthat is thermalizing or thermalized to a new distributiondue to the pump pulse Hence the thermoreectance of themetal dened as the change in reectance due to a change intemperature is given by

119889119889119889119889119889119889 10076491007649119879119879010076651007665

=119889119889excited minus 119889119889 10076491007649119879119879010076651007665

119889119889 10076491007649119879119879010076651007665 (2)

where 119889119889 is the reectivity 1198791198790 is the baseline temperatureof the sample and 119889119889excited is the reectivity of the metalin the state during and aer pump pulse absorption ethermoreectance of a metal is dictated by its thermore-ectance coecient 119889119889119889119889119889119889119889119879119879 which is typically small onthe order of 10minus6ndash10minus4 Kminus1 depending on the temperatureand metallic band structure at a given wavelength [58ndash60]erefore lock-in techniques are typically used to measurethe thermoreectance of metals in pump-probe geometriesUsing ultrashort laser pulses in pump-probe geometries thetime evolution of the thermoreectance can be measured[49 61]

e time evolution of the thermoreectance signal isthe basis behind time domain thermoreectance (TDTR)TDTR has been used in several different geometries withvarious different pulsed lasers ranging from nanoseconds tofemtoseconds [5 25 48 49 62ndash67] ese experiments canbe separated into two classes (1) experiments in which thelaser pulses are separated far enough in time in which theinterrogated sample is relaxed back down to its equilibriumstate before the next pulse arrives and (2) experiments inwhich the pulses are so closely spaced in time that thepulse arrives at the sample surface before full thermal relax-ation of the previous pulse Recently [68] the experimentsin the former class have been characterized as ldquotransientthermoreectance where experiments utilizing laser systemsproducing the later class of excitation have been commonlyreferred to as ldquotime domain thermoreectance (TDTR) InTDTR experiments the buildup of laser pulse energy leadsto a steady-state temperature rise in the sample [69] thatcan provide additional information for thermal analysisispaper will limit the review of measurement techniques toTDTR as the majority of the data presented in the latersections were measured with TDTR techniques

Due to the recent decrease in cost of femtosecondTi Sapphire oscillators most TDTR systems are centeredaround these high frequency (sim80MHz) laser systems Atypical cost of a Ti Sapphire oscillator from a commercialvendor is sim$100000ndash$200000 depending on the outputpower and pulse width e TDTR experiment that I havein my lab at University of Virginia is based off of a SpectraPhysics Tsunami with sim3W output of 90 fs laser pulsescentered at 800 nm is means that 90 fs laser pulses areemanating from the oscillator with about 375 J of energy

every 125 ns e pulse energy is split with a variable beamsplitter into a pump path and a probe path e pump pathis directed through an electro-optic modulator (EOM) thatcan modulate the pulse train upwards of 119891119891 = 119891119891MHz (Iuse the electro-optic modulators sold commercially throughConOptics) In my particular setup I then focus the modu-lated pumppulses through a bismuth triborate (BiBO) crystalto frequency double the near infrared pulses (800 nm =155 eV) to visible wavelengths (400 nm = 311 eV) I thendirect the pump pulses through an objective to focus the400 nm pump pulses onto the sample surfaceemodulatedpump pulses produce a temperature rise on the samplesurface at the frequency 119891119891 of the EOM

e pulse energy that is not directed down the pump pathaer the variable beam splitter is directed down amechanicaldelay stage and then focused collinearly with the pump pulsesthrough the objective these probe pulses are focused on thesample surface in the middle of the pump pulses I typicallymake the pump spot size larger than the probe spot size on thesample surface is reduces error in our measurements dueto probe beam dri and movement is also creates a one-dimensional heating event in the sample cross-plane to thesample surface (note if the pump beam waist is decreased tothe same spot size as the pump in-plane thermal transportproperties can also be measured however this is not ofinterest for this paper since I will only focus onmeasurementsof thermal boundary conductance across a thin interfacialregion where heat transfer is nearly entirely cross-plane) [6566] I typically use innity corrected objectiveswith anywherefrom 5timesndash20times magnication which leads to ranges of pumpand probe 1119889119890119890119891 radii anywhere from 25ndash6 120583120583m and 6ndash4 120583120583mrespectively depending on the objective Note that this is verydependent on the particular setup and the spot size of thepump and probe going into the back face of the objectivee reected probe beam is then reected along the sameincident path passed through polarizing and color lteringoptics and directed to a photodiode e voltage generatedfrom the photodiode contains both a DC signal and a smallAC component which represents the change in the probebeam reectivity due to the pump heating event at frequency119891119891 is AC component is ltered from the DC componentwith lock-in amplication and the in-phase and out-of-phasecomponents of the AC signal are monitored as a function ofpump-probe delay time which we can monitor over sim7 ns ofdelay time (although typically I nd that 4-5 ns is sucientto exact the thermal boundary conductance of the majorityof the samples of interest) A schematic of the TDTR systemin my lab at the University of Virginia is shown in Figure 3

TDTR data taken on a 117 nm Al lm evaporated ontoa single crystalline Si substrate are shown in Figure 4Characteristic in the TDTR data is a rapid rise in signalwhich is related to the increase in electronic temperaturedue to the pulse absorption followed by an exponentialdecay with different time constants related to the variouscooling processes in the Al lm In the rst sim10 ps thesignal decay is related to the hot electron equilibration withthe lattice indicative of short pulsed laser heating [49 6170ndash79] Once the electrons and phonon have equilibrated


SP Tsunami3 W 80 MHz

90 fs pluse width

Pump

Isolator

EOM

BiBO

Redfilter

Dichroic

Probe

Camera toimagesample

Blue

filter

Lock-inamplifier

Photodiode

Delay line (sim7 ns)

F 3 Schematic of TDTR system at the University of Virginia

0 1 2 3 4 5

0

1

2

3

4

5

6

7

8

9

10

Pump-probe time delay (ns)

TDTR data 115 nm AlSi

Thermal model

F 4 TDTR data on a 115 nm Al lm evaporated on a Sisubstrate along with the best t thermal model rior to Al evapora-tion the Si was processed with an acid etch to remove the majorityof the silicon native oxide layer e thermophysical propertiesdetermined from the model best t are ℎK = 380MWmminus2 Kminus1 forthe thermal boundary conductance across the AlSi interface and120581120581 = 11205811Wmminus2 Kminus1 for the thermal conductivity of the Si substrate

thermal diffusion through the Al lm occurs which typicallyoccurs on a time scale of 10ndash100 ps [48 80] From 100 psto several nanoseconds the TDTR signal is related to thethermal boundary conductance across the AlSi interfaceand the thermal effusivity or thermal diffusivity into the Sidepending on the modulation frequency of the pump [81]For purposes of measuring the thermal boundary conduc-tance between two solids the TDTR data between sim100 psto a few nanoseconds contains the pertinent information

To quantify the thermal boundary conductance fromthe TDTR data an accurate thermal model that relatesthe measured lock-in output to the thermal excitation andpropagation in the structures of interest must be used Formost TDTR experiments the use of Gaussian pump andprobe shapes along with a metal transducer that is thickerthan the optical penetration depths facilitates the solutionto the cylindrical heat equation in a multilayer structure (ametal lm that is sim80ndash120 nm should suffice) [69 82 83]e specics of the thermal model to predict the temperaturechange on the surface of the metal transducer have been

outlined in several works previously [65ndash67 69 84] It isimportant to note that due to the laser pulse accumulationin the sample from a Ti Sapphire oscillator the temperaturerise on the sample surface consists of two parts (1) a realcomponent that represents the temperature rise due to theexcitation of a single pulse and subsequent thermal diffusionfrom the surface of the metal into the underlying materialand across the material interfaces this real response isdirectly related to the time domain response of the surfacetemperature following the optical pump pulse and (2) animaginary component that represents the temperature risefrom the pulse accumulation this imaginary response is pre-dominantly controlled by the change in surface temperaturesubjected to the periodic heat source at frequency 119891119891 Boththe real and imaginary components of the heating event atthe sample surface contain information about the thermalproperties of the sample and can be used to analyze TDTRdata for robust data analysis e temperature rise predictedfrom the thermal model is related to the measured output ofthe lock-in amplier during TDTR experiments via

119881119881in = Re 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891

119881119881out = Im 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891(3)

where 119881119881in is the in-phase component of the lock-in transferfunction and related to the real component of the tempera-ture rise and119881119881out is the out-of-phase component of the lock-in transfer function and related to the imaginary componentof the temperature rise 119885119885119885119891119891119891 119891119891119885 which is the lock-in transferfunction is a function of pumpmodulation frequency119891119891 andsurface temperature rise 119891119891 and is given by

119885119885 10076491007649119891119891119891 11989111989110076651007665 = 120585120585infin10055761005576

119872119872=minusinfin119891119891 10076491007649119891119891 119891 119891119891laser10076651007665 exp 10077131007713120484120484119872119872119891119891laser12059112059110077291007729 119891 (4)

where 120585120585 is a constant related to the electronics thermore-ectance coefficient laser powers and frequency of the lasersource 119891119891laser and 120591120591 is the pump-probe delay time By natureof (4) pulse accumulation is taken into account Solving (3)for various pump-probe delay times 120591120591 yields the output ofthe lock-in amplier that is related to the measured datathrough the various thermal properties of the system Toanalyze the data the negative ratio of119881119881in to119881119881out119885minus119881119881in119881119881out119885yields a quantity that is directly related to the real temperaturerise on the surface of the sample and inversely related tothe imaginary temperature rise on the surface of the sampleUsing this ratio ismuchmore robust thanusing either119881119881in and119881119881out alone since it minimizes error due to probe beam driand dephasing decreases the noise due to the electronics andeliminates the need to scale the model to the experimentaldata or need to know the thermoreectance coefficient ofthe metal transducer [29 85] In Figure 4 I show the bestt of the model to the TDTR data along with the values ofAlSi ℎK and the Si 120581120581 that achieve this best t I oen use anSi substrate as a calibration of TDTR to ensure our experi-mental measurement can accurately reproduce the thermalconductivity of single crystalline silicon at room tempera-ture [86 87] (note substantially below room temperature


the thermal conductivity of single crystalline silicon cannotbe accurately measured due to ballistic phonon effects whichwill not be discussed in this paper however the reader isreferred to [88 89] that discussed this mean free path effectin more detail)

4 Semiclassical Formalism for PredictingThermal Boundary Conductance

Semiclassical modeling based on analytical expressions ofthe electron or phonon ux from a particle basis serves asa robust relatively simple approach to gain fundamentalinsight into the underlying process driving thermal bound-ary conductance across solid interfaces Although there aremore rigorous approaches to model the thermal boundaryconductance between two solids such asmolecular dynamicssimulations [21 90ndash95] or nonequilibrium Greenrsquos Functionmethods [96ndash100] due to their simplicity semiclassicalapproaches can be very powerful for quick predictions thatrequire very minimal computational expenses erefore Iwill restrict the discussion of thermal boundary conductancemodeling in this paper to semiclassical approaches anddraw on specic variations of these approaches for modelingimperfect interfaces in the sections that follow

e two most common approaches to modeling thermalboundary conductance are rooted in the assumptions ofeither a perfectly specular interface or a perfectly diffusiveinterface resulting in specular or diffuse scattering of theelectrons or phonons Typical approaches for modeling thespecular reection of electron or phonon waves at interfacesuse transmission calculations rooted in the Fresnel equa-tions [101] An example of this is the Acoustic MismatchModel for predicting the thermal boundary conductanceof phonons due to specular phonon wave interactions andtransmission at interfaces [102 103] ese approaches arerooted in the assumptions that the electrons or phonons willsee the interface as a perfectly specular wall From simpleanalytical arguments it is straight forward to show that thisassumption breaks down at temperatures above sim30K formostmaterials [104] even in the event of perfect interfaces aswas demonstrated experimentally [29] However in the eventthat imperfections or disorder are present at the interfacethis specular assumption breaks down even at temperaturesas low as 1K for certain interfaces [105] In this case theassumption of diffusive particle scattering must be employedto model the thermal transport across interfaces

e most common and straight forward approach formodeling diffusive phonon scattering across solid interfacesand the resulting thermal boundary conductance is imple-mentation of the Diffuse Mismatch Model (DMM) [17 105]e DMM is rooted in the fact that electrons or phononsscatter diffusively at the interface between two materialsand therefore the probability of the energy of these carriersto transmit energy from material 1 to material 2 is theprobability of energy reection from material 2 to material1 Mathematically this is expressed as

12057712057712 = 11990311990321 = 1 minus 12057712057721 (5)

where 120577120577 and 119903119903 are the probability of energy transmissionand reection respectively is diffusive transmission isthen multiplied by the phonon ux in side 2 incident onthe interface between material 1 and material 2 to give thetransmitted phonon ux from side 1 to side 2 which in thesemiclassical formalism is approximated by [16]

1199021199021 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895119891119891119891119891111989511989512057712057712119889119889120598120598 (6)

where 119895119895 is an index corresponding to the energy level (orphonon modebranch) 120598120598 is the energy of the carriers 119891119891 isthe statistical distribution function of the carriers 119891119891 is thevelocity of the carriers and 120598120598min and 120598120598max are the minimumand maximum energy of the carrier in level 119895119895 From (1)in the limit of an innitesimally small temperature rise thesemiclassical ldquoparticlerdquo formalism for the thermal boundaryconductance across interfaces is given by

ℎK12 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895

120597120597119891119891120597120597120597120597

119891119891111989511989512057712057712119889119889120598120598 (7)

which is applicable for both electrons and phonons (andphotons for that matter) More rigorous derivations of ℎK12are discussed elsewhere [95] In addition several works haveexamined the assumptions going into calculations of (7)including the use of phonondispersion [106] realistic densityof states [107] and isotropy [108ndash110] e calculation of12057712057712 relies on the denition of diffusive scattering (5) andthe fact that a diffusive scattering event at an interface willlead to a black thermalizing boundary In this case theprinciple of detailed balance can be evoked [111] and theDMM transmission is given by

12057712057712 =11990211990221

11990211990212 + 11990211990221 (8)

Note that this transmission expression is valid for bothelectrons and phonons (and again photons) scattering diffu-sively at interfaces I will not discuss the specics of the trans-mission calculations in this paper since the goal of this paperis to review recent advances in experimental observations ofthermal boundary conductance at rough interfaces Howeverseveral additional works have described the derivation ofthe DMM transmission coefficient in great detail and thereader is directed to the following references to understandthe fundamental assumptions of detailed balance and howit is applied to the DMM transmission [17 104] the role ofinelastic scattering on this energy transmission [112ndash114]and the role of uxes and transport around these interfaces[95 115 116] I will discuss the modications to (8) thathave beenmade to describe the additional processes affectingtransmission across imperfect interfaces in the sections thatfollow


5 Thermal Transport between TwoMetalsThe Electron-Dominated Thermal BoundaryConductance and the Effects ofCompositional Mixing

Typically the thermal boundary conductance in a system isonly a major concern when the interfacial transport is lowcompared to the transport in either of the materials compris-ing the interface However at metalmetal interfaces due tothe relatively high thermal conductivity of pure metals thetransport across these interfaces can be a limiting resistanceeven though the thermal boundary conductance can reachupwards of 10000MWmminus2 Kminus1 at room temperature

ere have only been limited experimental measure-ments of the thermal boundary conductance across metalmetal interfaces compared to those at interfaces comprised ofat least one nonmetalOne of the rst studies byClemens et al[117] reported on the thermal boundary conductance acrossa series of metalmetal bilayer and multilayer lms In thebilayer lms they were able to directly measure the thermalboundary conductance across NiTi and NiZr interfacesese values shown in Figure 5 are similar to the roomtemperature values of the TiNMgO interface [29] which asdiscussed previously is the highest phonon-dominated ther-mal boundary conductance ever measured Gundrum et al[28] measured the thermal boundary conductance acrossan interface between two free electron metals Al and Cue thermal boundary conductancewassim4000MWmminus2 Kminus1

at room temperature half-an-order of magnitude largerthan the highest phonon-dominated thermal boundary con-ductance ever measured is was theorized to be due topurely free-electron dominated transport across the AlCuinterfaces ey derived a variation of the DMM for freeelectrons given by

ℎK =1205741205741120592120592F11205741205742120592120592F2

4 100764910076491205741205741120592120592F1 + 1205741205742120592120592F210076651007665119879119879 (9)

where 120574120574 is the linear coefficient to the electron heat capacity(oen referred to as the Sommerfeld coefficient) [118] and120592120592F is the Fermi velocity (recently we have derived a moregeneral form of this electron DMM for nonfree electronmetals [119]) Calculations of (9) for an AlCu interfaceare shown as the line in Figure 5 which show acceptableagreement with the experimental data alluding to the factthat this interface is dominated by free electron transport

Recently Wilson and Cahill [27] followed up to thisAlCu measurement by measuring the thermal boundaryconductance deduced from thermal conductivity measure-ments of PdIr multilayers ese multilayers had very lowelectrical resistivity and the deduced thermal boundaryconductance is to date the highest thermal boundary con-ductance ever measured eir work also demonstrated thatthe Wiedemann-Franz Law [120] is valid at solid interfaces[121] further proving that the transport at metalmetalinterfaces is an electron-dominated phenomenone originof the low thermal boundary conductance of the NiTi andNiZr interfaces compared to the PdIr and AlCu interfacesis currently unknown however strong electron-phonon

75 100 200 300 400 500

Temperature (K)

50

100

1000

10000

(MW

mminus

2K

minus1)

DMMAlCu

AlCu

PdIr

NiZr

NiTi

TiNMgO

1014 1015 1016 101710

1000

10000

AlCu-785 K

AlCu-298 K

(MW

mminus

2K

minus1)

Ion dose (cmminus 2)

F 5ermal boundary conductance at metal-metal interfaces(PdIr [27] AlCu [28] NiTi [117] and NiZr [117]) compared tothe highest phonon-dominated thermal boundary conductance evermeasured (TiNMgO [29]) Inset thermal boundary conductance atthe AlCu interfaces as a function of ion dose

coupling effects in the Ni Ti and Zr transition metals whichwould lead to high electrical resistivities compared to Al Cuor the PdIr multilayers could be playing a role Howeverwith the recent result that theWiedemann-Franz Law is validat metalmetal interfaces [27] these underlying mechanismscould be deduced from electrical resistivity data

Due to this validity of theWiedemann-Franz law atmetal-metal interfaces disorder which would lead to an enhance-ment in electron-phonon and electron-defect scattering andthereby decrease the electrical conductivity (increase theelectrical resistivity) should also lead to a decrease in elec-tron thermal boundary conductance is was veried byGundrum et al [28] by ion irradiating the AlCu interfaceswith 1MeVKr ions to induce an AlCu intermixed layer atthe interface where the thickness of this layer would increasewith increasing ion dose As seen in the inset of Figure 5 thethermal boundary conductance across the AlCu interfacesystematically decreases with an increase in ion dose isis consistent with the increase in thickness of the AlCuintermixed layer which leads to increased electron scatteringaround the interface and therefore a decrease in the thermalboundary conductance from the Al to the Cu

In this section I reviewed the limited experimental dataon thermal boundary conductance at metalmetal interfacesand the effects of AlCu mixing at the interface leading to


a decrease in thermal boundary conductance In generalmetalmetal interfaces do not pose a limiting thermal resis-tance in most technology due to their relatively high ther-mal boundary conductances compared to metalnonmetalor nonmetalnonmetal interfaces where phonon scatteringdominates transport erefore for the remainder of thispaper I will focus on the effects of interfacial imperfectionson phonon transport by considering different types of imper-fections

6 Effects of Compositional Intermixing onPhonon Thermal Boundary Conductance

In the previous section I reviewed the effects of intermixingof Al and Cu on electron scattering and resulting thermalboundary conductance At the AlCu interface the thermalboundary conductance is dominated by electron transportHowever at interfaces in which one material is a nonmetalthe thermal boundary conductance is dominated by phonontransport [30]

We have studied the problem of compositional intermix-ing on phonon thermal boundary conductance bymeasuringℎK across CrSi interfaces with varying degrees of CrSiintermixed regions [22]We accomplished this via in situ pro-cessing of the Si surface prior to Cr lm sputtering followedby in situ heat treatments is led to compositionally mixedregions of CrSi around the CrSi interface that were mea-sured with Auger Electron Spectroscopy and TransmissionElectron Microscopy We measured the thermal boundaryconductance from the Cr to Si with a variation of TDTRwith a regeneratively amplied laser system (repetition rate250 kHz) [22 122] e measured thermal boundary con-ductance across these CrSi interfaces is shown in Figure 6e measured thermal boundary conductance decreases asthe compositionally mixed layer increases For comparisonI show the thermal boundary conductance from the AlCuinterfaces reported by Gundrum et al [28] and discussedin Section 5 e dependency of the AlCu electron domi-nated thermal boundary conductance as a function of layerthickness is similar to that at the CrSi phonon dominatedinterfacemdashthat is as the thickness of the compositionallymixed interfacial layer is increased the thermal boundaryconductance decreases

To understand the underlying phonon mechanisms driv-ing thermal boundary conductance at compositionallymixedinterfacial regions we extended the DMM for phonon trans-port outlined in Section 4 to a three-layer system We havediscussed this approach in detail in our previous works [123124] In short in this approach we described the interfacialcompositionally mixed region as a thin layer of thickness119889119889 with averaged properties of material 1 and material 2comprising the interface erefore for the CrSi interfacethe interfacial layer had averaged properties of Cr and Sie properties of this compositionally mixed interfacial layerwere calculated via the virtual crystal approximation [125]e thermal boundary conductance was then calculated by

ℎK12 = 10076501007650ℎminus1K1int + ℎ

minus1Kint210076661007666

minus1 (10)

0 2 4 6 8 10 12 14 16 18 20

Mixed layer thickness (nm)

100

1000

5000

AlCu

AlCu

(as received)

CrSi

VCDMM

approximation

to DMM

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)

F 6 ermal boundary conductance of a series of CrSiinterfaces as a function of CrSi mixed layer thickness [22] Forcomparison I also show the thermal boundary conductance fromthe AlCu interfaces reported by Gundrum et al [28] and discussedin Section 5 Compositional mixing leads to a decrease in bothelectron- and phonon-dominated thermal boundary conductance

where ldquointrdquo refers to the interfacial mixture the propertieswhich are described by a virtual crystal averaged property ofmaterial 1 and material 2 and ℎK is calculated via (7) etransmission coefficientwas then calculated via (8) where thephononuxes in each of thematerials and the interfacial layerare calculated via (6) As this model is based on the phononDMM we referred to this model as the Virtual Crystal DMM(VCDMM) Calculations of our VCDMM as a function ofof mixed layer thickness 119889119889 for a CrSi interface includinga varying width of a CrSi compositionally mixed region areshown in Figure 6 Details of these particular calculations arediscussed in our previous work [22] e model shows verygood agreement with our experimentally measured data

e calculations discussed above for the VCDMM forthe CrSi interface were specic for phonon interactionsand scattering in a compositionally disordered CrSi layerof varying thickness 119889119889 Inherent to these calculations werethe virtual crystal assumption of averaged layer propertiesbased on a model derived for crystalline alloys [125] eproperties of an interfacial mixed layer can take on a range ofstructures andmorphologies and depending on thematerialscomprising the interface the region may or may not form analloy compoundWe developed an extension to the VCDMMto account for complete disorder in an intermixed region

10 ISRN Mechanical Engineering

(ie amorphous) [124] however in general these approach-es must be further validated With the limited experimentaldata on the effects of compositional mixing on thermalboundary conductance it is difficult to draw any concreteconclusions on either the fundamental electron or phononmechanisms driving transport in an interfacial mixed regionHowever it is apparent that the presence of this composi-tionally mixed interfacial region will decrease the interfacialthermal transport from one material to another

7 Effects of Roughness on Phonon ThermalBoundary Conductance

e previous section discussed the effects of compositionalmixing on phonon-mediated thermal boundary conduc-tance In this case the interface is not compositionallyabrupt but consists of some region of graded compositionalproperties that was referred to as a mixed layer is begsthe question what type of thermal perturbations and phononscattering mechanisms can exist at compositional abruptyet nonperfect interfaces We studied this problem by con-sidering thermal transport across interfaces between thinAl lms and geometrically roughened Si substrates Withpre-Al-deposition Si processing the silicon surface createda geometrically rough boundary between the Al and SiHowever by ensuring that the native oxide layer reformson the Si aer processing the Al will not migrate or spikeinto the Si as the native oxide layer in the Si acts as adiffusion barrier is creates a compositionally abrupt yetgeometrically rough AlSi interface

In a recent series of papers [19ndash21] we explored theextent ofmanipulation of phonon thermal boundary conduc-tance through either chemical roughening of the Si [20 21]or creating roughness by synthesizing Si1minus119909119909Ge119909119909 quantumdotson an Si surface In both situations the samples were exposedto ambient atmosphere to allow the formation of a silicaor germania native oxide layer aer surface modication toserve as a diffusion barrier to the Al transducer as mentionedabove this was conrmed with cross-sectional transmissionelectron microscopy [21] We quantied the roughness ofthe various surfaces with atomic force microscopy and usedthe root mean square (RMS) roughness of the silicon surface(120575120575) as the metric for comparison We measured the thermalboundary conductance across the AlSi interfaces from 100to 300K with TDTR

e thermal boundary conductance as a function oftemperature for the Alchemically roughened Si interfaces[21] is shown in Figure 7 (lled symbols) In addition Ihave included the Kapitza conductance at a nominally atand oxide-free AlSi interface as reported in [89] (opensquares) As the data indicate even a thin native oxide layerat the interface substantially reduces the effective Kapitzaconductance (gt50 reduction at room temperature) Inaddition these two datasets demonstrate signicantly dif-ferent temperature dependencies suggesting that the oxidelayer inhibitsmultiple-phonon scattering eventswhichwouldotherwise contribute to Kapitza conductance [112 113]Similarly comparing the four datasets of the present study

50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )

F 7 Predicted and measured values of Kapitza conductanceat AlSi interfaces plotted as a function of temperature e opencircles are the measured values at oxide-free AlSi interfaces from[89] and the lled symbols are the data measured in our previouswork [21] It is evident that both the presence of a native oxidelayer and interface roughness can have a signicant effect onKapitza conductance Not only does roughness decrease Kapitzaconductance but also it suppresses the temperature dependencee agreement between the dashdot lines and the data suggests thatthe DMM can be adjusted to take into account both the presence ofan oxide layer and interface roughness

increased interface roughness both reduces the magnitudeof Kapitza conductance as well as suppresses its temperaturedependencemdashthat is Kapitza conductance is less tempera-ture dependent as interface roughness increases

In addition to the data I have plotted several differentpredictive models in Figure 7 All models are calculatedassuming that elastic phonon-phonon interactions domi-nate Kapitza conductancemdashthat is phonons in silicon atfrequencies higher than the maximum phonon frequencyof aluminum do not participate in transport e diffusemismatch model [17] is calculated using assumptions thatwe have previously outlined in [106] where the vibrationalproperties of lm and substrate are approximated by ttingpolynomials to the phonon dispersion curves of aluminum[126] and silicon [127] along the [100] crystallographic direc-tion assumed spherical Brillouin zones are then constructedvia an isotropic revolution of these polynomial ts in wave-vector space As seen in Figure 7 the prediction of the DMMfalls between the data of [89] and that at the smoothest


interface presently considered is suggests that without anoxide layer inelastic phonon-phonon scattering could play arole in thermal transport across aluminum-silicon interfaces[112 113] On the other hand we attribute the differencebetween the predicted and measured values at the smoothestinterface considered (black squares) to the native oxide layere conductance of this oxide layer is described by its thermalconductivity divided by its thickness

ℎoxide =120581120581oxide119905119905oxide

(11)

When evaluating (12) we use the temperature-dependentbulk thermal conductivity of 120572120572SiO2 as it has been shownthat the thermal conductivity of thin lm 120572120572SiO2 does notsubstantially differ from that of bulk [29] A series resistorapproach then yields

ℎK = 10076501007650ℎminus1KDMM + ℎminus1oxide10076661007666

minus1 (12)

is prediction is represented by the solid black line in Figure7 and agrees well with our experimental data

In order to take interfacial roughness into account weintroduce a spectral attenuation coefficient (discussed indetail in our previous work [19]) and insert this coefficientinto the integral expression of the DMM is coefficient 120574120574is unity when the phonon wavelength 120582120582 is greater than theRMS roughness 120575120575 On the other hand 120574120574 = 120574120574120574120574minus120574120574120574120574120574120574120574120582120582120574120575120575120574when 120582120582 120582 120575120575 at is phonons with wavelengths greaterthan 120575120575 are unaffected by the roughness of the interfacewhereas those with wavelengths less than 120575120575 are affected in afashion similar to that of photons in an absorptive media forexample the Beer-Lambert law Qualitatively speaking thisapproach suggests that as the ldquoabsorptivityrdquo of the interfaceincreases so too does the temperature drop across it Withthe spectral attenuation coefficient implemented the DMMis once again plotted in Figure 7 for roughnesses of 65 nmand 114 nm As is evident in the plot this approach notonly captures the reduction in Kapitza conductance dueto interface roughening but also captures the reduction intemperature dependence as well

Finally I plot room temperature Kapitza conductance as afunction of RMS roughness in Figure 8 (squares) comparingthe chemically roughened data from our paper [21] theaforementioned roughness DMMmodel calculated at 300Kand two prior sets of experimental data from our previousworks in which we explored other methods of rougheningthe silicon surface (different chemical etchants (le pointingtriangles) [20] and quantum dot roughening (circles) [19])as mentioned at the beginning of this section Regardlessof the silicon roughening procedure the thermal boundaryconductances all follows a similar relationship with RMSroughness

To summarize we have extensively characterized rough-ness effects on thermal boundary conductance across AlSiinterfaces In general we found that an increase in RMSroughness leads to a systematic decrease in thermal boundaryconductance We developed an extension to the DMMthat is based around phonon wavelength considerations in

005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225

250DMM + oxide+ roughness

Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)

F 8 Room-temperature predicted (line) and measured ther-mal boundary conductance at AlSi interfaces plotted as a functionof interface roughness Regardless of the silicon roughening proce-dure (indicated in the plot) the thermal boundary conductances allfollows a similar relationship with RMS roughness

which only certain phonon wavelengths see the interfacialroughness is model captures both the magnitude of theexperimental data and the temperature and roughness trendswell However this RMS roughness phenomenon to myknowledge has only been intricately explored with well-characterized interfaces in our three recent works [19ndash21]Different materials with different phonon spectra and oxidereactivities (eg gold which does not wet a Si native oxidelayer) should be explored to generalize the phonon physics atgeometrically rough interfaces

8 Effects of Lattice Dislocations on PhononThermal Boundary Conductance

us far I have discussed the effects of compositionalmixing and geometric roughness on thermal transport acrosssolid interfaces ese interfacial characteristics can com-monly be produced by nanoscale processing or modifyingthe growthsynthesis conditions However combinations ofmaterials can lead to additional interfacial imperfectionsbased on the crystal structure and atomic properties emost common imperfections that naturally form at the inter-face between two solids are dislocations produced from strainrelaxation that stem from the lattice mismatch between twomaterials Dislocations which can readily scatter phononsand add to thermal resistance in solids [128] have beenshown to cause drastic reductions in the thermal conductivityof thin lms and nanostructures For example at largeperiodic thicknesses (sim10 nm) the thermal conductivity ofSiGe superlattices has been shown to suddenly drop bya factor of two due to dislocation formation from strainrelaxation of the Si or Ge layers on Ge or Si respectivelyleading to poor crystalline quality [129] Similarly threadingdislocation densities of 120574 times 1010 cmminus1 have led to AlNlms on Si substrates to exhibit thermal conductivities


less than the theoretical minimum limit [130 131] Asthese dislocations stem from interfacial growth and latticerelaxation understanding how dislocations affect thermalboundary conductance adds an important piece to the studyof the phononmechanisms affecting thermal transport acrossimperfect interfaces

is issue was addressed in part by Costescu et al [29]who measured the thermal conductivity across TiNAl2O3interfaces in addition to TiNMgO interfaces (one of theTiNMgO samples that was measured is shown in Figure 2)e measured TiNMgO (001) which is essentially free ofmist dislocations TiNMgO (111) which contains a highdensity of stacking faults but few mist dislocations andTiNAl2O3 which should contain a large density of bothstacking faults and mist dislocations show no appreciabledifference in the thermal boundary conductance is indi-cates that in these well acoustically matched materials thedislocation densities did not affect the phonon transportHowever without quantifying the dislocation density it isdifficult to understand the effects of these dislocations onthermal boundary conductance In addition without quan-tication this does not add any direct insight into previousstudies observing drastically reduced thermal conductivitiesof thin lms due to dislocation formation (such as the AlNthin lm example above [130])

We quantitatively studied the effects of dislocation den-sity on thermal boundary conductance by measuring thethermal boundary conductance across well-characterizedGaSbGaAs interfaces as is outlined in detail in our previouswork [23] e lattice mismatch between GaSb and GaAsis 778 and the critical thickness for strain relaxation isformed within the rst monolayer Different epitaxial growthmodes were utilized to change the dislocation densities at theGaSbGaAs interfaces as characterized with plan view andcross-sectional transmission electron microscopy [132 133]ese growth techniques were able to produce GaSbGaAsinterfaces with two ranges of dislocations densities 5 times 106ndash5 times 108 dislocations cmminus2 and 109ndash1011 dislocations cmminus2We measured the thermal boundary conductances across theGaSbGaAs interfaces from 100ndash500K with TDTR

e thermal boundary conductances as a function oftemperature for the two GaSbGaAs interfaces with differentdislocation densities are shown in Figure 9 (lled and opentriangles for the relatively low and high dislocation densesamples resp) It is interesting to note that roughly a twoorder of magnitude change in dislocation density only leadsto a factor of two change in thermal boundary conductanceTo put this into perspective at similar dislocation levelsfor two orders of magnitude change in dislocation densitythe thermal conductivity of GaN drops by an order ofmagnitude [130] is indicates the seemingly small effectthat changes in dislocation densities have on thermal bound-ary conductance across heavily dislocated interfaces Forcomparison I also plot the thermal boundary conductanceof TiNMgO [29] and Bidiamond [30] interfaces (as dis-cussed in Section 2 TiNMgO and Bidiamond representthe highest and lowest phonon-mediated thermal boundaryconductances ever measured) e GaSbGaAs dislocation

75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)

(5 times 106ndash5 times 108 dislocations cmminus 2)

(109ndash1011 dislocations cmminus 2)

F 9 ermal boundary conductances at the two differentdislocation dense GaSbGaAs interfaces from our previous work[23] ese interfaces have dislocation densities ranging from 5 times106ndash5 times 108 dislocations cmminus2 and 109ndash1011 dislocations cmminus2 Forcomparison the plot includes previouslymeasured data at TiNMgO[29] and Bidiamond [30] interfaces DMM calculations are shownfor the GaSbGaAs interfaces which drastically overpredict themeasured data since the traditional formulation of the DMM doesnot account for dislocations interfaces It is also interesting tonote that roughly a two order of magnitude change in dislocationdensity only leads to a factor of two change in thermal boundaryconductance

dense interfaces are approaching the measured values ofBidiamond However Bi and diamond have drasticallydifferent phonon spectra and are extremely acousticallymismatched (over an order of magnitude different acousticvelocities and Debye Temperatures) [134] Based on DMMtheory acoustically mismatched interfaces will have lowerphonon transmission probability than acoustically matchedinterfaces [17] GaSb and GaAs have similar phonon spectraand phonon velocities and are well acoustically matchede TiNMgO interfaces are well acoustically matched andwere epitaxially grown (like the GaSb) e differences inthermal boundary conductances between the GaSbGaAsand TiNMgO well acoustically matched interfaces are overan order of magnitude indicating the potential strong effectthat dislocations can have on phonon transmission isis further exemplied by comparing our measured thermalboundary conductances across the GaSbGaAs interfaces toDMM calculations for GaSbGaAs For these calculations

ISRN Mechanical Engineering 13

we used the approach that we outlined previously in whichwe t a polynomial to a measured or simulated phonondispersion in one crystallographic direction and assume anisotropic medium to calculate the DMM [106] For DMMcalculations we used the dispersion in the Γ rarr X directionof the Brillouin zone from [135] for GaSb and [136] forGaAs (we include the optical branches of GaSb and GaAsin our calculations) [137] I plot these calculations of theDMM forGaSbGaAs as the solid line in Figure 9 Clearly theDMMgreatly overpredicts themeasured values As theDMMassumes a perfect interface and a single-phonon scatteringevent this approach is not valid for the dislocation denseGaSbGaAs interfaces

e discussion in this section demonstrates the detri-mental effects that dislocations can have on thermal bound-ary conductance However due to the limited data thefundamental mechanisms driving phonon dislocation inter-actions around interfaces are unknown One outstandingquestion is the functional form of the dependency of thermalboundary conductance on dislocation density In the ther-mal conductivity of a homogeneous material the thermalconductivity is inversely proportional to the square root ofthe dislocation density [130] is implies that changes indislocation densities at a dislocation light interface will begreater than at a dislocation dense interface Assuming theGaSbGaAs interface is in the dislocation dense regime thiscould explain the relatively small change in thermal boundaryconductance (factor of two) with a two order of magnitudechange in dislocation density However as discussed byCostescu et al [29] there is essentially no difference inthermal boundary conductance between TiNMgO (001)TiNMgO (111) and TiNAl2O3 which has varying levels ofdislocation densities is implies that either the dislocationdensities although varying were extremely low at these TiN-based interfaces however without exact characterization ofthe defect densities at these interfaces quantitative analysesare not possible erefore to elucidate the underlying roleof dislocations on phonon thermal boundary conductancecontrolled experiments of thin lms on substrates withvarying degrees of dislocation formation at the lm substrateboundaries need to be pursued

9 Effect of Bonding on Phonon ThermalBoundary Conductance

us far I have discussed the roles that imperfections in thelattice structure or order of themasses around solid interfaceshave on thermal boundary conductance However phonontransport is also affected by the strength of the interatomicbondinge question then arises what effect does interfacialbond strength have on phonon thermal boundary conduc-tance Very recently this has been an active area of researchin nanoscale heat transport Several works have investigatedthis problem computationally or theoretically [92 93 138ndash142] Experimentally the effects of bond strength on thermalboundary conductance have been pin-pointed in a few spe-cic systems using self-assembledmonolayers diamond anviltechnologies or surface termination to tune the interfacial

interactions Hsieh et al [143] used a diamond anvil cellto vary the pressure at AlSiC interfaces and demonstratedan increase in thermal boundary conductance with pressurewhen an acoustically so layer is placed between the Aland SiC (such as graphene which interacts with the Alvia van der Waal interactions or SiO119909119909 which has as lowacoustic impedance) e stiffness of these ldquoweakrdquo interfacesincreased with pressure and a nearly order of magnitudeincrease in thermal boundary conductance between the Aland SiC was demonstrated at pressures of sim10GPa comparedto no applied pressure Losego et al [144] tuned the bondingbetween Au and quartz by using self-assembled monolay-ers (SAMs) sandwiched at the interface e end groupchemistries of the SAMs covalently bonded to the quartzgave a method of tuning the SAMAu interaction strengthis led to tunability in the thermal transport from the Auto the quartz by increasing the density of covalent bondsthat bridge the Au to the substrate Collins and Chen [145]and Monachon and Weber [146] demonstrated the use ofsurface termination of diamond as a means to increasing thethermal boundary conductance between various metals anddiamond substrates Most notably they found that oxygentermination can increase the thermal boundary conductancebetweenAl and diamond by a factor of two (Collins andChen[145]) and Cr and diamond by a factor of 11 (Monachon andWeber [146]) at room temperature as compared to untreateddiamond

We investigated this problem by utilizing adsorbed ele-mental functional groups at graphene contacts [24] As previ-ously mentioned graphene interfaces are typically describedas weakly interacting due to the van der Waal forces thatare characteristic of cross plane graphenegraphite Using lowpressure electron beam generated plasmas [147 148] a seriesof graphene lms on SiO2Si substrates were functionalizedwith O2 Aer plasma functionalization the graphene sur-faces were coated with Al and we measured the thermalboundary conductance between the Al and graphene I showthe thermal boundary conductance between Alsingle layergraphene (SLG) and Aloxygen functionalized SLG (AlO-SLG) in Figure 10 We observed a factor of two increasein ℎK across the AlO-SLG interface as compared with theAlSLG interfaceWe attributed this behavior to the enhancedreactivity of the oxygenated surface which is consistent withthe increase observed across the AlO-diamond interface ascompared to the Aldiamond interface [145 149] also shownin Figure 10 e presence of oxygen-containing functionalgroups increases the surface energy and thusmakes graphenemore reactive allowing a lower energy bonding state with theevaporated Al lm as compared to the untreated SLG whichis chemically inertis lower energy bonding state promotesAl-O formation leading to a strongerAl-O-SLG junction andthereby an increase in ℎK

To quantify this we turn to the DMM discussed inSection 4 However in this case the transmission coefficientmust be rederived to account for the two-dimensional natureof the graphene To calculate the transmission coefficientwe used the assumption that we discussed in a previouswork [110] in which we treat the graphene sheet as a two-dimensional solid thereby performing detailed balance on


50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)

F 10 ermal boundary conductance across our AlSLG(lled triangles) and AlO-SLG (unlled triangles) interfaces [24]e thermal transport across the AlSLG interface increases bya factor of two with oxygen functionalization of the grapheneOur measured values are in good agreement with conductancesmeasured across Aldiamond and AlO-diamond interfaces whichexhibit an increase in thermal boundary conductance with oxygentermination [145 149] We model the thermal conductance acrossthe AlSLG interface with the DMM as shown by the solid line Weadjust the velocity of the SLG in our DMMcalculations tomodel theAlO-SLG thermal boundary conductance (dashed line) and ndthat the resultant velocity in the oxygenated SLG is about a factor oftwo higher than the non-functionalized sampleis is indicative ofthe increase in bond strength between the Al and the SLG via theoxygen adsorbates

the uxes in the Al and SLG For the ux in the Al we canmake the usual three-dimensional isotropic approximationof phonon ux discussed in Section 4 However given thetwo-dimensional nature of the SLG the phonon ux in thismaterial is given by [110]

119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)

where 119886119886 is the interlayer spacing of graphite (ie = 335Aring)[150] For calculations of this anisotropic SLG-modiedDMM (which we refer to in Figure 10 as ldquoDMM-2D varia-tionrdquo) we t polynomials to the phonon dispersions of Alin the Γ rarr X direction [126] and of graphene in theΓ rarr K direction [151] to give greater accuracy in the DMMcalculations as compared to the Debye approximation [106]

We note however that the modes in the Γ rarr K directionin graphene are extremely Debye-like (ie non dispersive)in the frequency regime that is elastically accessible to the Al(frequencies below 10THz) We ignore the ZA mode in thegraphene dispersion since the ZAmode is heavily suppressedin encased and supported SLG [43 152] Given that ourmeasurements represented the cross plane conductance ofAlSLG the cross plane velocity of SLG is meaninglesserefore we dene 119891119891SLG as a tting parameter that weassume is constant with frequency is is a similar approachas performed in the analysis of Koh et al [153] in whichthey adjust the transmission coecient to t the DMM totheir AuTiSLG data and Schmidt et al [154] in whichthey adjust the phonon transmission to t the DMM totheir metalgraphite data We note that this approach ofadjusting only 119891119891SLG gives us more direct insight into howthe bonding at the AlSLG interface changes due to thefunctionalization since we are not adjusting any aspect ofthe graphene dispersion only the transport velocity ets of the DMM are shown in Figure 10 e solid line isthe DMM calculation assuming the graphene cross planevelocities are 2455 and 1480m sminus1) for the longitudinal andtransverse modes of the SLG respectively (labeled ldquoDMM-2D variationrdquo) e dashed line is the DMM calculationassuming 4687 and 2825m sminus1 for the longitudinal andtransverse modes of the O-SLG respectively e velocitiesthat result in the best t of the DMM to the SLG data are ingood agreement with the cross plane velocity of bulk graphite[150] which could be indicative of similar bonding betweenthe Al and SLG as the van der Waals bonds cross plane ingraphite e velocities resulting in the best t in the AlO-SLG data are nearly a factor of two higher which we attributeto the increased bond strength between the Al and SLGdue to the presence of oxygen leading to covalent bondingbetween the Al and O-SLG It is important to note that weuse the velocity parameter and tting only to lend insightinto the nature of the bonding at the two different interfacesSince the velocity is proportional to the square root of thespring constant the increase by a factor of two in the best-tvelocity with oxygen functionalization implies that increasein thermal transport is due to an increase in interfacial bondstrength by roughly 40

In summary in this section I have discussed the roleof interfacial bonding on thermal boundary conductanceat solid interfaces is has been a topic of several recentworks as outlined in detailed in this section e strengthof the interfacial bond can be manipulated to directly tunethe interfacial conductance offering a unique ldquoknobrdquo forthermal transport In our specic work outlined in [24]we showed that the conductance of a single layer grapheneinterface can be manipulated by elemental adsorbates fromlow energy plasma functionalization It is interesting to notethat the temperature dependency of the thermal bound-ary conductance also changes which could be evidence ofhigher order phonon interactions such as 3- or 4-phononconversion processes at the interface [138] However theexact phonon conversion mechanisms occurring at variablybonded interfaces are unclear and more research must be


pursued to elucidate the fundamental phonon interactionproblem

10 Summary

e bottle neck of many modern technologies and energysolutions has boiled down to thermal engineering problemon the nanoscale As both increases and decreases in thermalconductivity are required for different applications the abilityto control and tune the thermal properties of materials istherefore the ultimate goal to improve both device efficiencyand energy consumption trendsis desired thermal controlin nanosystems however is not a trivial task Due to themagnitudes of the thermal mean free paths approachingor overpassing typical length scales in nanomaterials (iematerials with length scales less than 1 micron) the thermaltransport across interfaces can dictate the overall resistancein nanosystems However the fundamental mechanismsdriving these electron and phonon interactions at nanoscaleinterfaces are difficult to predict and control since thethermal boundary conductance across interfaces (oen calledthe Kapitza conductance) ℎK is intimately related to thecharacteristics of the interface (structure bonding geometryetc) in addition to the fundamental atomistic properties ofthe materials comprising the interface itself is interplaybetween interfacial properties and thermal boundary con-ductance can lead to additional carrier scattering mecha-nisms and phenomena that will not only change the thermalprocesses but can be used to systematically manipulate andcontrol thermal transport in nanosystems

In this paper I have reviewed recent experimentalprogress in understanding this interplay between interfacialproperties on the atomic scale and thermal transport acrosssolid interfaces I focused this discussion specically on therole of interfacial nanoscale ldquoimperfectionsrdquo such as surfaceroughness compositional disorder atomic dislocations orinterfacial bonding In general these interfacial imperfec-tions lead to decreases in thermal boundary conductancewhere each type of imperfection leads to different scatteringmechanisms that can be used to control the thermal bound-ary conductance is offers a unique avenue for controllingscattering and thermal transport in nanotechnology

Acknowledgments

e author is grateful to the support from the NationalScience Foundation (CBET Grant no 1134311) and SandiaNational Laboratories through the LDRD program office

References

[1] Lawrence Livermore National Laboratory Livermore CalifUSA 94550 LLNL Energy Flow Chart httpsowchartsllnlgov

[2] ldquoEPA report on clean energy and air emissions [online]rdquohttpwwwepagovcleanenergyenergy-and-youaffectair-emissionshtml

[3] G E Moore ldquoCramming more components onto integratedcircuitsrdquo Electronics Magazine vol 38 p 4 1965

[4] E Pop ldquoEnergy dissipation and transport in nanoscale devicesrdquoNano Research vol 3 no 3 pp 147ndash169 2010

[5] D G Cahill W K Ford K E Goodson et al ldquoNanoscalethermal transportrdquo Journal of Applied Physics vol 93 no 2 pp793ndash818 2003

[6] T Hendricks and W T Choate ldquoEngineering scoping studyof thermoelectric generator systems for industrial waste heatrecoveryrdquo Tech Rep Pacic Northwest National Laboratoryand BCS 2006

[7] K Yazawa and A Shakouri ldquoCost-efficiency trade-off andthe design of thermoelectric power generatorsrdquo EnvironmentalScience and Technology vol 45 pp 7548ndash7553 2011

[8] M S Dresselhaus G Dresselhaus X Sun et al ldquoe promiseof low-dimensional thermoelectric materialsrdquoMicroscale er-mophysical Engineering vol 3 no 2 pp 89ndash100 1999

[9] B K Nayak V V Iyengar and M C Gupta ldquoEfficient lighttrapping in silicon solar cells by ultrafast-laser-induced self-assembled micronano structuresrdquo Progress in Photovoltaicsvol 19 pp 631ndash639 2011

[10] T Kietzke ldquoRecent advances in organic solar cellsrdquo Advances inOptoElectronics vol 2007 Article ID 40285 15 pages 2007

[11] A N Smith and J P Calame ldquoImpact of thin lm thermophys-ical properties on thermal management of wide bandgap solid-state transistorsrdquo International Journal ofermophysics vol 25no 2 pp 409ndash422 2004

[12] G J Snyder and E S Toberer ldquoComplex thermoelectricmaterialsrdquo Nature Materials vol 7 no 2 pp 105ndash114 2008

[13] B Poudel Q Hao Y Ma et al ldquoHigh-thermoelectric per-formance of nanostructured bismuth antimony telluride bulkalloysrdquo Science vol 320 no 5876 pp 634ndash638 2008

[14] A I Hochbaum R Chen R D Delgado et al ldquoEnhanced ther-moelectric performance of rough silicon nanowiresrdquo Naturevol 451 no 7175 pp 163ndash167 2008

[15] A I Boukai Y Bunimovich J Tahir-Kheli J K Yu W AGoddard and J R Heath ldquoSilicon nanowires as efficient ther-moelectric materialsrdquo Nature vol 451 no 7175 pp 168ndash1712008

[16] G Chen Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005

[17] E T Swartz and R O Pohl ldquoermal boundary resistancerdquoReviews of Modern Physics vol 61 no 3 pp 605ndash668 1989

[18] P L Kapitza ldquoe study of heat transfer in Helium IIrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki p 1 1941

[19] P E Hopkins J C Duda C W Petz and J A Floro ldquoControl-ling thermal conductance through quantum dot roughening atinterfacesrdquo Physical Review B vol 84 no 3 Article ID 0354387 pages 2011

[20] P E Hopkins L M Phinney J R Serrano and T E BeechemldquoEffects of surface roughness and oxide layer on the thermalboundary conductance at aluminumsilicon interfacesrdquo Phys-ical Review B vol 82 no 8 Article ID 085307 5 pages 2010

[21] J C Duda and P E Hopkins ldquoSystematically controllingKapitza conductance via chemical etchingrdquo Applied PhysicsLetters vol 100 no 11 Article ID 111602 4 pages 2012

[22] P E Hopkins P M Norris R J Stevens T E Beechem and SGraham ldquoInuence of interfacial mixing on thermal boundaryconductance across a chromiumsilicon interfacerdquo Journal ofHeat Transfer vol 130 no 6 Article ID 062401 10 pages 2008

[23] P E Hopkins J C Duda S P Clark et al ldquoEffect of dislocationdensity on thermal boundary conductance across GaSbGaAs


interfacesrdquo Applied Physics Letters vol 98 no 16 Article ID161913 3 pages 2011

[24] P E Hopkins M Baraket E V Barnat et al ldquoManipulatingthermal conductance at metal-graphene contacts via chemicalfunctionalizationrdquo Nano Letters vol 12 pp 590ndash595 2012

[25] D G Cahill K Goodson and A Majumdar ldquoermometryand thermal transport in micronanoscale solid-state devicesand structuresrdquo Journal of Heat Transfer vol 124 no 2 pp223ndash241 2002

[26] D G Cahill ldquoExtremes of heat conduction Pushing theboundaries of the thermal conductivity of materialsrdquo MRSBulletin vol 37 pp 855ndash863 2012

[27] R B Wilson and D G Cahill ldquoExperimental validation of theinterfacial form of the wiedemann-franz lawrdquo Physical ReviewLetters vol 108 Article ID 255901 5 pages 2012

[28] B C Gundrum D G Cahill and R S Averback ldquoermalconductance of metal-metal interfacesrdquo Physical Review B vol72 no 24 5 pages 2005

[29] R M Costescu M A Wall and D G Cahill ldquoermalconductance of epitaxial interfacesrdquo Physical Review B vol 67no 5 Article ID 054302 5 pages 2003

[30] H K Lyeo andDG Cahill ldquoermal conductance of interfacesbetween highly dissimilar materialsrdquo Physical Review B vol 73no 14 Article ID 144301 6 pages 2006

[31] D G Cahill ldquoermal conductivity measurement from 30 to750 K the 3120596120596methodrdquo Review of Scientic Instruments vol 61no 2 article 802 7 pages 1990

[32] Y C Tai CHMastrangelo andR SMuller ldquoermal conduc-tivity of heavily doped low-pressure chemical vapor depositedpolycrystalline silicon lmsrdquo Journal of Applied Physics vol 63no 5 article 1442 6 pages 1988

[33] L M Phinney E S Piekos and J D Kuppers ldquoBond padeffects on steady state thermal conductivity measurement usingsuspended micromachined test structuresrdquo in Proceedings ofthe ASME International Mechanical Engineering Congress andExposition (IMECE rsquo07) vol 41349 Seattle Wash USA 2007

[34] S Uma A D McConnell M Asheghi K Kurabayashi and KE Goodson ldquoTemperature-dependent thermal conductivity ofundoped polycrystalline silicon layersrdquo International Journal ofermophysics vol 22 no 2 pp 605ndash616 2001

[35] A D McConnell S Uma and K E Goodson ldquoermalconductivity of doped polysilicon layersrdquo Journal of Microelec-tromechanical Systems vol 10 no 3 pp 360ndash369 2001

[36] D Li Y Wu P Kim L Shi P Yang and A Majumdarldquoermal conductivity of individual silicon nanowiresrdquoAppliedPhysics Letters vol 83 no 14 article 2934 3 pages 2003

[37] Z Chen W Jang W Bao C N Lau and C Dames ldquoer-mal contact resistance between graphene and silicon dioxiderdquoApplied Physics Letters vol 95 no 16 Article ID 161910 3pages 2009

[38] L Lu W Yi and D L Zhang ldquo3 120596120596 method for specic heatand thermal conductivity measurementsrdquo Review of ScienticInstruments vol 72 no 7 pp 2996ndash3003 2001

[39] T Tong and A Majumdar ldquoReexamining the 3-omega tech-nique for thin lm thermal characterizationrdquo Review of Scien-tic Instruments vol 77 no 10 Article ID 104902 9 pages2006

[40] M L Bauer C M Bauer M C Fish et al ldquoin-lm aerogelthermal conductivity measurements via 3120596120596rdquo Journal of Non-Crystalline Solids vol 357 no 15 pp 2960ndash2965 2011

[41] B W Olson S Graham and K Chen ldquoA practical extensionof the 3120596120596 method to multilayer structuresrdquo Review of ScienticInstruments vol 76 no 5 Article ID 053901 7 pages 2005

[42] P E Hopkins and L M Phinney ldquoermal conductivitymeasurements on polycrystalline silicon microbridges usingthe 3120596120596 techniquerdquo Journal of Heat Transfer vol 131 no 4Article ID 043201 8 pages 2009

[43] J H Seol I Jo A L Moore et al ldquoTwo-dimensional phonontransport in supported graphenerdquo Science vol 328 no 5975pp 213ndash216 2010

[44] L Shi D Li C Yu et al ldquoMeasuring thermal and thermoelectricproperties of one-dimensional nanostructures using a micro-fabricated devicerdquo Journal of Heat Transfer vol 125 no 5 pp881ndash888 2003

[45] H S Carslaw and J C Jaeger Conduction of Heat in SolidsOxford University Press New York NY US 2nd edition 1959

[46] H W Deem and W D Wood ldquoFlash thermal-diffusivitymeasurements using a laserrdquo Review of Scientic Instrumentsvol 33 no 10 pp 1107ndash1109 1962

[47] J Guo X Wang and T Wang ldquoermal characterization ofmicroscale conductive and nonconductive wires using transientelectrothermal techniquerdquo Journal of Applied Physics vol 101no 6 Article ID 063537 7 pages 2007

[48] C A Paddock and G L Eesley ldquoTransient thermoreectancefrom thin metal lmsrdquo Journal of Applied Physics vol 60 no 1pp 285ndash290 1986

[49] G L Eesley ldquoObservation of nonequilibrium electron heat-ing in copperrdquo Physical Review Letters vol 51 no 23 pp2140ndash2143 1983

[50] J H Weaver D W Lynch C H Culp and R Rosei ldquoer-moreectance of V Nb and paramagnetic Crrdquo Physical ReviewB vol 14 no 2 pp 459ndash463 1976

[51] E Colavita A Franciosi DW Lynch G Paolucci andR Roseildquoermoreectance investigation of the antiferromagnetic andparamagnetic phases of Crrdquo Physical Review B vol 27 no 3 pp1653ndash1663 1983

[52] E Colavita A Franciosi C Mariani and R Rosei ldquoermore-ectance test of W Mo and paramagnetic Cr band structuresrdquoPhysical Review B vol 27 no 8 pp 4684ndash4693 1983

[53] D W Lynch R Rosei and J H Weaver ldquoInfrared andvisible optical properties of single crystal Ni at 4Krdquo Solid StateCommunications vol 9 no 24 pp 2195ndash2199 1971

[54] R Rosei and D W Lynch ldquoermomodulation spectra of AlAu and Curdquo Physical Review B vol 5 no 10 pp 3883ndash38941972

[55] R Rosei C H Culp and J H Weaver ldquoTemperature modula-tion of the optical transitions involving the fermi surface in Agexperimentalrdquo Physical Review B vol 10 no 2 pp 484ndash4891974

[56] R Rosei ldquoTemperature modulation of the optical transitionsinvolving the fermi surface in Ag theoryrdquo Physical Review Bvol 10 no 2 pp 474ndash483 1974

[57] J L Hostetler A N Smith and P M Norris ldquoin-lm ther-mal conductivity and thickness measurements using picosec-ond ultrasonicsrdquoMicroscale ermophysical Engineering vol 1no 3 pp 237ndash244 1997

[58] W P Hsieh and D G Cahill ldquoTa and Au(Pd) alloy metallm transducers for time-domain thermoreectance at highpressuresrdquo Journal of Applied Physics vol 109 no 11 ArticleID 113520 4 pages 2011


[59] Y Wang J Y Park Y K Koh and D G Cahill ldquoer-moreectance of metal transducers for time-domain thermore-ectancerdquo Journal of Applied Physics vol 108 no 4 Article ID043507 4 pages 2010

[60] P E Hopkins ldquoEffects of electron-boundary scattering onchanges in thermoreectance in thin metal lms undergoingintraband excitationsrdquo Journal of Applied Physics vol 105 no9 Article ID 093517 6 pages 2009

[61] G L Eesley ldquoGeneration of nonequilibrium electron and latticetemperatures in copper by picosecond laser pulsesrdquo PhysicalReview B vol 33 no 4 pp 2144ndash2151 1986

[62] P M Norris A P Caffrey R J Stevens J M Klopf JT McLeskey and A N Smith ldquoFemtosecond pump-probenondestructive examination of materials (invited)rdquo Review ofScientic Instruments vol 74 no 1 pp 400ndash406 2003

[63] M N Touzelbaev P Zhou R Venkatasubramanian andK E Goodson ldquoermal characterization of Bi2Te3Sb2Te3superlatticesrdquo Journal of Applied Physics vol 90 no 2 pp763ndash767 2001

[64] P M Norris and P E Hopkins ldquoExamining interfacial diffusephonon scattering through transient thermoreectance mea-surements of thermal boundary conductancerdquo Journal of HeatTransfer vol 131 no 4 Article ID 043207 11 pages 2009

[65] A J Schmidt X Chen and G Chen ldquoPulse accumulationradial heat conduction and anisotropic thermal conductivity inpump-probe transient thermoreectancerdquo Review of ScienticInstruments vol 79 no 11 Article ID 114902 9 pages 2008

[66] P E Hopkins J R Serrano L M Phinney S P Kearney TW Grasser and C omas Harris ldquoCriteria for cross-planedominated thermal transport in multilayer thin lm systemsduring modulated laser heatingrdquo Journal of Heat Transfer vol132 no 8 Article ID 081302 10 pages 2010

[67] P E Hopkins B Kaehr L M Phinney et al ldquoMeasuring thethermal conductivity of porous transparent SiO2 lms withtime domain thermoreectancerdquo Journal of Heat Transfer vol133 no 6 Article ID 061601 8 pages 2011

[68] Y K Koh S L Singer W Kim et al ldquoComparison of the 3120596120596method and time-domain thermoreectance formeasurementsof the cross-plane thermal conductivity of epitaxial semicon-ductorsrdquo Journal of Applied Physics vol 105 no 5 Article ID054303 7 pages 2009

[69] D G Cahill ldquoAnalysis of heat ow in layered structures fortime-domain thermoreectancerdquo Review of Scientic Instruments vol 75 no 12 pp 5119ndash5122 2004

[70] S I Anisimov B L Kapeliovich and T L Perelman ldquoElec-tron emission from metal surfaces exposed to ultrashort laserpulsesrdquo Soviet Physics vol 39 pp 375ndash377 1974

[71] P E Hopkins J M Klopf and P M Norris ldquoInuence of inter-band transitions on electron-phonon coupling measurementsin Ni lmsrdquoApplied Optics vol 46 no 11 pp 2076ndash2083 2007

[72] P E Hopkins L M Phinney and J R Serrano ldquoRe-examiningelectron-fermi relaxation in gold lms with a nonlinear ther-moreectance modelrdquo Journal of Heat Transfer vol 133 no 4Article ID 044505 4 pages 2011

[73] P E Hopkins J L Kassebaum and P M Norris ldquoEffects ofelectron scattering at metal-nonmetal interfaces on electron-phonon equilibration in gold lmsrdquo Journal of Applied Physicsvol 105 no 2 Article ID 023710 8 pages 2009

[74] P E Hopkins and PM Norris ldquoSubstrate inuence in electron-phonon coupling measurements in thin Au lmsrdquo AppliedSurface Science vol 253 no 15 pp 6289ndash6294 2007

[75] P E Hopkins ldquoInuence of inter- and intraband transitions toelectron temperature decay in noble metals aer short-pulsedlaser heatingrdquo Journal of Heat Transfer vol 132 no 12 ArticleID 122402 6 pages 2010

[76] P E Hopkins ldquoContributions of inter- and intraband excita-tions to electron heat capacity and electron-phonon coupling innoblemetalsrdquo Journal of Heat Transfer vol 132 no 1 Article ID014504 4 pages 2010

[77] J Hohlfeld S S Wellershoff J Guumldde U Conrad V Jaumlhnkeand E Matthias ldquoElectron and lattice dynamics followingoptical excitation of metalsrdquoChemical Physics vol 251 no 1ndash3pp 237ndash258 2000

[78] T Q Qiu and C L Tien ldquoHeat transfer mechanisms duringshort-pulse laser heating of metalsrdquo Journal of Heat Transfervol 115 no 4 article 835 7 pages 1993

[79] T Q Qiu and C L Tien ldquoSize effects on nonequilibrium laserheating of metal lmsrdquo Journal of Heat Transfer vol 115 no 4article 842 6 pages 1993

[80] A Caffrey P Hopkins J Klopf and P Norris ldquoin lm non-noble transition metal thermophysical propertiesrdquo Nanoscaleand Microscale ermophysical Engineering vol 9 no 4 pp365ndash377 2005

[81] A J Schmidt R Cheaito andM Chiesa ldquoA frequency-domainthermoreectance method for the characterization of thermalpropertiesrdquo Review of Scientic Instruments vol 80 no 9Article ID 094901 6 pages 2009

[82] H S Carslaw and J C Jaeger ldquoSteady periodic temperature incomposite slabsrdquo in Conduction of Heat in Solids pp 109ndash112OxfordUniversity PressNewYorkNYUSA 2nd edition 2003

[83] A Feldman ldquoAlgorithm for solutions of the thermal diffusionequation in a stratied medium with a modulated heatingsourcerdquo High Temperatures vol 31 no 3 pp 293ndash296 1999

[84] A Schmidt M Chiesa X Chen and G Chen ldquoAn opticalpump-probe technique for measuring the thermal conductivityof liquidsrdquoReview of Scientic Instruments vol 79 no 6 ArticleID 064902 5 pages 2008

[85] D G Cahill and FWatanabe ldquoermal conductivity of isotopi-cally pure and Ge-doped Si epitaxial layers from 300 to 550 KrdquoPhysical Review B vol 70 no 23 Article ID 235322 3 pages2004

[86] F Incropera and D P DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons New York NY USA 4th edition1996

[87] D R Lide CRC Handbook for Chemistry and Physics Taylor ampFrancis Boca Raton Fla USA 89th edition 2008

[88] Y K Koh and D G Cahill ldquoFrequency dependence of thethermal conductivity of semiconductor alloysrdquo Physical ReviewB vol 76 no 7 Article ID 075207 5 pages 2007

[89] A J Minnich J A Johnson A J Schmidt et al ldquoermalconductivity spectroscopy technique to measure phonon meanfree pathsrdquo Physical Review Letters vol 107 no 9 Article ID095901 4 pages 2011

[90] R J Stevens L V Zhigilei and P M Norris ldquoEffects oftemperature and disorder on thermal boundary conductanceat solid-solid interfaces nonequilibrium molecular dynamicssimulationsrdquo International Journal of Heat and Mass Transfervol 50 no 19-20 pp 3977ndash3989 2007

[91] J CDuda T S English E S Piekos T E BeechemTWKennyand P E Hopkins ldquoBidirectionally tuning Kapitza conductancethrough the inclusion of substitutional impurities rdquo Journal ofApplied Physics vol 112 no 7 Article ID 073519 p 5 2012


[92] Z Y Ong and E Pop ldquoFrequency and polarization dependenceof thermal coupling between carbon nanotubes and SiO2rdquoJournal of Applied Physics vol 108 no 10 Article ID 1035028 pages 2010

[93] Z Y Ong and E Pop ldquoMolecular dynamics simulation ofthermal boundary conductance between carbon nanotubes andSiO2rdquo Physical Review B vol 81 no 15 Article ID 155408 7pages 2010

[94] E S Landry and A J H McGaughey ldquoEffect of interfacialspecies mixing on phonon transport in semiconductor super-latticesrdquo Physical Review B vol 79 no 7 Article ID 075316 8pages 2009

[95] E S Landry and A J H McGaughey ldquoermal boundaryresistance predictions from molecular dynamics simulationsand theoretical calculationsrdquo Physical Review B vol 80 no 16Article ID 165304 11 pages 2009

[96] Z Huang J Y Murthy and T S Fisher ldquoModeling ofpolarization-specic phonon transmission through interfacesrdquoJournal of Heat Transfer vol 133 Article ID 114502 3 pages2011

[97] W Zhang T S Fisher and N Mingo ldquoSimulation of interfacialphonon transport in Si-Ge heterostructures using an atomisticgreenrsquos function methodrdquo Journal of Heat Transfer vol 129 no4 pp 483ndash491 2007

[98] P E Hopkins P M Norris M S Tsegaye and A W GhoshldquoExtracting phonon thermal conductance across atomic junc-tions nonequilibrium greenrsquos function approach compared tosemiclassical methodsrdquo Journal of Applied Physics vol 106 no6 Article ID 063503 10 pages 2009

[99] N Mingo inermal Nanosystems and Nanomaterials S VolzEd vol 118 of Topic in Applied Physics Springer BerlinGermany 2009

[100] W Zhang T S Fisher and N Mingo ldquoe atomisticGreenrsquos function method an efficient simulation approach fornanoscale phonon transportrdquo Numerical Heat Transfer B vol51 no 4 pp 333ndash349 2007

[101] G Laufer Introduction to Optics and Lasers in EngineeringCambridge University Press Cambridge UK 1996

[102] W A Little ldquoe transport of heat between dissimilar solids atlow temperaturesrdquo Canadian Journal of Physics vol 37 no 3pp 334ndash349 1959

[103] I M Khalatnikov and I N Adamenko ldquoeory of the Kapitzatemperature discontinuity at solid body-liquid helium bound-aryrdquo Soviet Physics vol 36 p 391 1973

[104] J C Duda P E Hopkins J L Smoyer et al ldquoOn the assumptionof detailed balance in prediction of diffusive transmission prob-ability during interfacial transportrdquo Nanoscale and Microscaleermophysical Engineering vol 14 no 1 pp 21ndash33 2010

[105] E T Swartz and R O Pohl ldquoermal resistance at interfacesrdquoApplied Physics Letters vol 51 no 26 pp 2200ndash2202 1987

[106] J C Duda T E Beechem J L Smoyer P M Norris and P EHopkins ldquoRole of dispersion on phononic thermal boundaryconductancerdquo Journal of Applied Physics vol 108 no 7 ArticleID 073515 10 pages 2010

[107] P Reddy K Castelino and A Majumdar ldquoDiffuse mismatchmodel of thermal boundary conductance using exact phonondispersionrdquo Applied Physics Letters vol 87 no 21 Article ID211908 pp 1ndash3 2005

[108] P E Hopkins T E Beechem J C Duda et al ldquoInuence ofanisotropy on thermal boundary conductance at solid inter-facesrdquo Physical Review B vol 84 no 12 Article ID 125408 7pages 2011

[109] R S Prasher ldquoermal boundary resistance and thermalconductivity ofmultiwalled carbon nanotubesrdquo Physical ReviewB vol 77 no 7 Article ID 075424 11 pages 2008

[110] J C Duda J L Smoyer P M Norris and P E HopkinsldquoExtension of the diffuse mismatch model for thermal bound-ary conductance between isotropic and anisotropic materialsrdquoApplied Physics Letters vol 95 no 3 Article ID 031912 3 pages2009

[111] W G Vincenti and C H Kruger Introduction to Physical GasDynamics Krieger Publishing Company Malabar Fla USA2002

[112] P E Hopkins ldquoMultiple phonon processes contributing toinelastic scattering during thermal boundary conductance atsolid interfacesrdquo Journal of Applied Physics vol 106 no 1Article ID 013528 9 pages 2009

[113] P E Hopkins J C Duda and P M Norris ldquoAnharmonicphonon interactions at interfaces and contributions to thermalboundary conductancerdquo Journal of Heat Transfer vol 133Article ID 062401 11 pages 2011

[114] C Dames and G Chen ldquoeoretical phonon thermal con-ductivity of SiGe superlattice nanowiresrdquo Journal of AppliedPhysics vol 95 no 2 pp 682ndash693 2004

[115] G Chen ldquoDiffusion-transmission interface condition for elec-tron and phonon transportrdquo Applied Physics Letters vol 82 no6 pp 991ndash993 2003

[116] S Simons ldquoOn the thermal contact resistance between insu-latorsrdquo Journal of Physics C vol 7 no 22 article 009 pp4048ndash4052 1974

[117] BM Clemens G L Eesley and C A Paddock ldquoTime-resolvedthermal transport in compositionally modulated metal lmsrdquoPhysical Review B vol 37 no 3 pp 1085ndash1096 1988

[118] N W Ashcro and N D Mermin Solid State Physics SaundersCollege 1976

[119] P E Hopkins T E Beechem J C Duda J L Smoyer and P MNorris ldquoEffects of subconduction band excitations on thermalconductance at metal-metal interfacesrdquo Applied Physics Lettersvol 96 no 1 Article ID 011907 3 pages 2010

[120] G Wiedemann and R Franz ldquoUeber die warme-leitungsfa-higkeit der Metallerdquo Annalen Der Physik vol 165 no 4 pp978ndash531 1853

[121] G D Mahan and M Bartkowiak ldquoWiedemann-Franz law atboundariesrdquo Applied Physics Letters vol 74 no 7 pp 953ndash9541999

[122] P E Hopkins R J Stevens and P M Norris ldquoInuence ofinelastic scattering at metal-dielectric interfacesrdquo Journal ofHeat Transfer vol 130 no 2 Article ID 022401 9 pages 2008

[123] T Beechem S Graham P Hopkins and P Norris ldquoRole ofinterface disorder on thermal boundary conductance using avirtual crystal approachrdquo Applied Physics Letters vol 90 no 5Article ID 054104 2007

[124] T Beechem and P E Hopkins ldquoPredictions of thermalboundary conductance for systems of disordered solids andinterfacesrdquo Journal of Applied Physics vol 106 no 12 ArticleID 124301 8 pages 2009

[125] B Abeles ldquoLattice thermal conductivity of disordered semicon-ductor alloys at high temperaturesrdquo Physical Review vol 131no 5 pp 1906ndash1911 1963

[126] G Gilat and R M Nicklow ldquoNormal vibrations in aluminumand derived thermodynamic propertiesrdquo Physical Review vol143 no 2 pp 487ndash494 1966


[127] R Weber ldquoMagnon-phonon coupling in metallic lmsrdquo Physi-cal Review vol 169 no 2 pp 451ndash456 1968

[128] G P Srivastavae Physics of Phonons Taylor amp Francis NewYork NY USA 1990

[129] S M Lee D G Cahill and R Venkatasubramanian ldquoermalconductivity of Si-Ge superlatticesrdquo Applied Physics Letters vol70 no 22 pp 2957ndash2959 1997

[130] Z Su L Huang F Liu et al ldquoLayer-by-layer thermal conduc-tivities of theGroup III nitride lms in bluegreen light emittingdiodesrdquo Applied Physics Letters vol 100 no 20 Article ID201106 4 pages 2012

[131] D G Cahill S K Watson and R O Pohl ldquoLower limit to thethermal conductivity of disordered crystalsrdquo Physical Review Bvol 46 no 10 pp 6131ndash6140 1992

[132] A Jallipalli G Balakrishnan S H Huang et al ldquoStructuralanalysis of highly relaxed GaSb grown on GaAs substrates withperiodic interfacial array of 90∘ mist dislocationsrdquo NanoscaleResearch Letters vol 4 no 12 pp 1458ndash1462 2009

[133] S H Huang G Balakrishnan A Khoshakhlagh A Jallipalli LR Dawson and D L Huffaker ldquoStrain relief by periodic mistarrays for low defect density GaSb on GaAsrdquo Applied PhysicsLetters vol 88 no 13 Article ID 131911 3 pages 2006

[134] D E Gray American Institute of Physics Handbook McGrawHill New York NY USA 3rd edition 1972

[135] M K Farr J G Traylor and S K Sinha ldquoLattice dynamics ofGaSbrdquo Physical Review B vol 11 no 4 pp 1587ndash1594 1975

[136] K Kunc and R M Martin ldquoAb initio force constants ofgaas a new approach to calculation of phonons and dielectricpropertiesrdquo Physical Review Letters vol 48 no 6 pp 406ndash4091982

[137] T Beechem J C Duda P E Hopkins and P M NorrisldquoContribution of optical phonons to thermal boundary conduc-tancerdquo Applied Physics Letters vol 97 no 6 Article ID 0619073 pages 2010

[138] J C Duda T S English E S Piekos W A Soffa L V ZhigileiandP EHopkins ldquoImplications of cross-species interactions onthe temperature dependence of Kapitza conductancerdquo PhysicalReview B vol 84 no 19 Article ID 193301 4 pages 2011

[139] M Shen W J Evans D Cahill and P Keblinski ldquoBondingand pressure-tunable interfacial thermal conductancerdquo PhysicalReview B vol 84 no 19 Article ID 195432 6 pages 2011

[140] L Zhang P Keblinski J SWang and B Li ldquoInterfacial thermaltransport in atomic junctionsrdquo Physical Review B vol 83 no 6Article ID 064303 9 pages 2011

[141] L Hu L Zhang M Hu J S Wang L Baowen and P KeblinskildquoPhonon interference at self-assembled monolayer interfacesmolecular dynamics simulationsrdquo vol 81 no 23 Article ID235427 5 pages 2010

[142] M Hu P Keblinski and P K Schelling ldquoKapitza conductanceof silicon-amorphous polyethylene interfaces by moleculardynamics simulationsrdquoPhysical ReviewB vol 79 no 10 ArticleID 104305 7 pages 2009

[143] W P Hsieh A S Lyons E Pop P Keblinski and D GCahill ldquoPressure tuning of the thermal conductance of weakinterfacesrdquo Physical Review B vol 84 no 18 Article ID 1841075 pages 2011

[144] M D Losego M E Grady N R Sottos D G Cahill and PV Braun ldquoEffects of chemical bonding on heat transport acrossinterfacesrdquo Nature Materials vol 11 pp 502ndash506 2012

[145] K C Collins and G Chen ldquoEffects of surface chemistryon thermal conductance at aluminum-diamond interfacesrdquo

Applied Physics Letters vol 97 no 8 Article ID 083102 3 pages2010

[146] C Monachon and L Weber ldquoermal boundary conductanceof transition metals on diamondrdquo Emerging Materials Researchvol 1 pp 89ndash98 2012

[147] M Baraket S G Walton E H Lock J T Robinson and FK Perkins ldquoe functionalization of graphene using electron-beam generated plasmasrdquoApplied Physics Letters vol 96 no 23Article ID 231501 3 pages 2010

[148] E H Lock M Baraket M Laskoski et al ldquoHigh-qualityuniform dry transfer of graphene to polymersrdquo Nano Lettersvol 12 pp 102ndash107 2012

[149] R J Stoner H J Maris T R Anthony and W F BanholzerldquoMeasurements of the Kapitza conductance between diamondand several metalsrdquo Physical Review Letters vol 68 no 10 pp1563ndash1566 1992

[150] R Nicklow N Wakabayashi and H G Smith ldquoLatticedynamics of pyrolytic graphiterdquo Physical Review B vol 5 pp4951ndash4962 1972

[151] S V Kusminskiy D K Campbell and A H C Neto ldquoLenoskyrsquosenergy and the phonondispersion of graphenerdquoPhysical ReviewB vol 80 no 3 Article ID 035401 3 pages 2009

[152] W Jang Z ChenW Bao C N Lau and C Dames ldquoickness-dependent thermal conductivity of encased graphene andultrathin graphiterdquoNano Letters vol 10 no 10 pp 3909ndash39132010

[153] Y K KohMH Bae D G Cahill and E Pop ldquoHeat conductionacross monolayer and few-layer graphenesrdquo Nano Letters vol10 no 11 pp 4363ndash4368 2010

[154] A J Schmidt K C Collins A J Minnich and G Chenldquoermal conductance and phonon transmissivity of metal-graphite interfacesrdquo Journal of Applied Physics vol 107 no 10Article ID 104907 5 pages 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



the drive of the computing industry to maintain Moorersquos Law[3] and recently focused on mitigating the thermal transportin the nanoscale devices [4 5] Recent novel approachesin reducing the net wasted heat have focused on wasteheat recycling via thermoelectric power generation [6ndash8]ermoelectric solutions also show promise in reducingharmful emissions in the grid as the entirely solid statemakeup of thermoelectrics requires no moving parts cyclesor input fuel other than heat ese thermoelectric solutionscombined with ldquorenewable sourcerdquo technologies (technolo-gies that do not rely on coal as a fuelmdashsuch as photovoltaics[9 10]) could substantially reduce the carbon footprint of theenergy grid

e issue with these technologies lies in their operationefficiencies which at this stage of research have boileddown to a thermal engineering problem on the nanoscaleFor example power dissipation limits the performances ofelectronic systems from individual microprocessors to datacenters In silicon-based microprocessors (by far the mostcommonly implemented technology) operation beyond afew GHz is not possible due to on-chip power densitiesexceeding 100Wcmminus2 higher than a typical hotplate andfar greater than typical cooling capabilities [4] Even greaterpower densities have been observed in RF and amplierdevices using IIIndashV or group IV compound semiconductors(eg GaAs or SiC) [11] ese huge power densities leadto hot-spots and device failure warranting the need forincreased thermal conduction to mitigate the heat load Asanother example current thermoelectric solutions for wasteheat recovery are far outperformed by traditional solutionssuch as heat exchangers with heat pumps or heat enginecycles such as the Organic Rankine or Kalina cycles [6] mak-ing major implementations of the more-carbon-footprint-friendly thermoelectric devices impractical Improvement inthe thermoelectric efficiency can be achieved by controlledreduction in thermal transport of the thermoelectric mate-rials since the thermoelectric efficiency (quantied by theFigure of Merit ZT) is inversely related to the thermalconductivity Recentmassive improvements in ZT in thermo-electric materials have been discovered by nanostructuring[8 12ndash15] leaving the research problem to improve energyefficiency centered around understanding thermal transporton the nanoscale As both increases and decreases in thermalconductivity are required for different applications the abilityto control and tune the thermal properties of materials istherefore the ultimate goal to improve device efficiency andimprove energy consumption trends

is desired thermal control in nanosystems howeveris not a trivial task Due to the magnitudes of the thermalmean free paths approaching or overpassing typical lengthscales in nanomaterials (ie materials with length scales lessthan 1 micron) the thermal transport across interfaces candictate the overall resistance in nanosystems is leads to aregime in which unique yet relatively unknown mechanismssuch as interfacial scattering ballistic transport and wave-effects offer potential new degrees of freedom in the ther-mal engineering of material systems [5 16] However thefundamental mechanisms driving these electron and phonon

Perfect

interface

Substrate

Film

Imperfect

interface

Film

Substrate

F 1 Schematic of a perfect and an atomically imperfectinterface A perfect interface is an atomically abrupt coherentinterface between two solids in intimate contact e atomicallyimperfect interface depicted here occurs when the interface isnot well dened due to atomic disorder (both structural andcompositional) resulting in roughness and impurities around theinterface

interactions at nanoscale interfaces are difficult to predictand control since the thermal boundary conductance acrossinterfaces (oen called the Kapitza conductance) ℎK [17 18]is intimately related to the characteristics of the interface(structure bonding geometry etc) in addition to the funda-mental atomistic properties of the materials comprising theinterface itself is interplay between interfacial propertiesand thermal boundary conductance can lead to additionalcarrier scattering mechanisms and phenomena that will notonly change the thermal processes but can also be used tosystematically manipulate and control thermal transport innanosystems

As an example consider a ldquoperfectrdquo interface and anatomically disordered ldquoimperfectrdquo interface depicted inFigure 1e imperfect interface introduces roughness alongwith atomic imperfections that cause additional thermal scat-tering which can change ℎK as I have discussed in previousworks [19ndash24] and will detail throughout this paper eproperties of this interface dictate the additional scatteringmechanisms erefore an understanding of the effects ofinterfacial imperfections and other ldquonon-idealitiesrdquo (suchas roughness disorder and bonding) on thermal transportacross complex nanoscale interfaces is warranted to fullydevelop the potential to control electron and phonon thermalprocesses in nanosystems

To this end in this paper I will review recent advance-ments in the underlying physics of thermal boundary con-ductance at solid interfaces focusing specically on therole of interfacial nanoscale ldquoimperfectionsrdquo such as surfaceroughness compositional disorder atomic dislocations orinterfacial bonding In the next section I present an overviewof thermal boundary conductance anddiscuss pertinentmea-surements of thermal boundary conductance between twosolids to put this thermophysical quantity into perspectiveFollowing this background I discuss time domain thermore-ectance in Section 3 TDTRhas become a common effectiveexperimental technique to accurately measure the thermalboundary conductance between two solids In Section 4





ℎK =119902119902intΔ119879119879

(1)




75 100 200 300 400 500

Temperature (K)

1

10

100

1000

10000= 01 nm

PdIr

AlCu


AlSLG

(bonding)= 10 nm


Bidiamond

= 100 nm = 1000 nmTh

erm

al b

ou

nd

ary

con

du

ctan

ce

(MW

mminus

2K

minus1) = 1 nm




=120581120581SiO2













119889119889119889119889119889119889 10076491007649119879119879010076651007665

=119889119889excited minus 119889119889 10076491007649119879119879010076651007665

119889119889 10076491007649119879119879010076651007665 (2)









90 fs pluse width

Pump

Isolator

EOM

BiBO

Redfilter

Dichroic

Probe


Blue

filter

Lock-inamplifier

Photodiode



0 1 2 3 4 5

0

1

2

3

4

5

6

7

8

9

10



Thermal model





119881119881in = Re 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891

119881119881out = Im 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891(3)


119885119885 10076491007649119891119891119891 11989111989110076651007665 = 120585120585infin10055761005576









12057712057712 = 11990311990321 = 1 minus 12057712057721 (5)


1199021199021 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895119891119891119891119891111989511989512057712057712119889119889120598120598 (6)


ℎK12 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895

120597120597119891119891120597120597120597120597

119891119891111989511989512057712057712119889119889120598120598 (7)


12057712057712 =11990211990221

11990211990212 + 11990211990221 (8)







ℎK =1205741205741120592120592F11205741205742120592120592F2

4 100764910076491205741205741120592120592F1 + 1205741205742120592120592F210076651007665119879119879 (9)



75 100 200 300 400 500

Temperature (K)

50

100

1000

10000

(MW

mminus

2K

minus1)

DMMAlCu

AlCu

PdIr

NiZr

NiTi

TiNMgO

1014 1015 1016 101710

1000

10000

AlCu-785 K

AlCu-298 K

(MW

mminus

2K

minus1)












ℎK12 = 10076501007650ℎminus1K1int + ℎ

minus1Kint210076661007666

minus1 (10)

0 2 4 6 8 10 12 14 16 18 20


100

1000

5000

AlCu

AlCu

(as received)

CrSi

VCDMM

approximation

to DMM

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )







(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













ℎK =119902119902intΔ119879119879

(1)




75 100 200 300 400 500

Temperature (K)

1

10

100

1000

10000= 01 nm

PdIr

AlCu


AlSLG

(bonding)= 10 nm


Bidiamond

= 100 nm = 1000 nmTh

erm

al b

ou

nd

ary

con

du

ctan

ce

(MW

mminus

2K

minus1) = 1 nm




=120581120581SiO2













119889119889119889119889119889119889 10076491007649119879119879010076651007665

=119889119889excited minus 119889119889 10076491007649119879119879010076651007665

119889119889 10076491007649119879119879010076651007665 (2)









90 fs pluse width

Pump

Isolator

EOM

BiBO

Redfilter

Dichroic

Probe


Blue

filter

Lock-inamplifier

Photodiode



0 1 2 3 4 5

0

1

2

3

4

5

6

7

8

9

10



Thermal model





119881119881in = Re 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891

119881119881out = Im 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891(3)


119885119885 10076491007649119891119891119891 11989111989110076651007665 = 120585120585infin10055761005576









12057712057712 = 11990311990321 = 1 minus 12057712057721 (5)


1199021199021 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895119891119891119891119891111989511989512057712057712119889119889120598120598 (6)


ℎK12 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895

120597120597119891119891120597120597120597120597

119891119891111989511989512057712057712119889119889120598120598 (7)


12057712057712 =11990211990221

11990211990212 + 11990211990221 (8)







ℎK =1205741205741120592120592F11205741205742120592120592F2

4 100764910076491205741205741120592120592F1 + 1205741205742120592120592F210076651007665119879119879 (9)



75 100 200 300 400 500

Temperature (K)

50

100

1000

10000

(MW

mminus

2K

minus1)

DMMAlCu

AlCu

PdIr

NiZr

NiTi

TiNMgO

1014 1015 1016 101710

1000

10000

AlCu-785 K

AlCu-298 K

(MW

mminus

2K

minus1)












ℎK12 = 10076501007650ℎminus1K1int + ℎ

minus1Kint210076661007666

minus1 (10)

0 2 4 6 8 10 12 14 16 18 20


100

1000

5000

AlCu

AlCu

(as received)

CrSi

VCDMM

approximation

to DMM

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )







(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















119889119889119889119889119889119889 10076491007649119879119879010076651007665

=119889119889excited minus 119889119889 10076491007649119879119879010076651007665

119889119889 10076491007649119879119879010076651007665 (2)









90 fs pluse width

Pump

Isolator

EOM

BiBO

Redfilter

Dichroic

Probe


Blue

filter

Lock-inamplifier

Photodiode



0 1 2 3 4 5

0

1

2

3

4

5

6

7

8

9

10



Thermal model





119881119881in = Re 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891

119881119881out = Im 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891(3)


119885119885 10076491007649119891119891119891 11989111989110076651007665 = 120585120585infin10055761005576









12057712057712 = 11990311990321 = 1 minus 12057712057721 (5)


1199021199021 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895119891119891119891119891111989511989512057712057712119889119889120598120598 (6)


ℎK12 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895

120597120597119891119891120597120597120597120597

119891119891111989511989512057712057712119889119889120598120598 (7)


12057712057712 =11990211990221

11990211990212 + 11990211990221 (8)







ℎK =1205741205741120592120592F11205741205742120592120592F2

4 100764910076491205741205741120592120592F1 + 1205741205742120592120592F210076651007665119879119879 (9)



75 100 200 300 400 500

Temperature (K)

50

100

1000

10000

(MW

mminus

2K

minus1)

DMMAlCu

AlCu

PdIr

NiZr

NiTi

TiNMgO

1014 1015 1016 101710

1000

10000

AlCu-785 K

AlCu-298 K

(MW

mminus

2K

minus1)












ℎK12 = 10076501007650ℎminus1K1int + ℎ

minus1Kint210076661007666

minus1 (10)

0 2 4 6 8 10 12 14 16 18 20


100

1000

5000

AlCu

AlCu

(as received)

CrSi

VCDMM

approximation

to DMM

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )







(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











90 fs pluse width

Pump

Isolator

EOM

BiBO

Redfilter

Dichroic

Probe


Blue

filter

Lock-inamplifier

Photodiode



0 1 2 3 4 5

0

1

2

3

4

5

6

7

8

9

10



Thermal model





119881119881in = Re 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891

119881119881out = Im 10077131007713119885119885 10076491007649119891119891119891 1198911198911007665100766510077291007729 119891(3)


119885119885 10076491007649119891119891119891 11989111989110076651007665 = 120585120585infin10055761005576









12057712057712 = 11990311990321 = 1 minus 12057712057721 (5)


1199021199021 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895119891119891119891119891111989511989512057712057712119889119889120598120598 (6)


ℎK12 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895

120597120597119891119891120597120597120597120597

119891119891111989511989512057712057712119889119889120598120598 (7)


12057712057712 =11990211990221

11990211990212 + 11990211990221 (8)







ℎK =1205741205741120592120592F11205741205742120592120592F2

4 100764910076491205741205741120592120592F1 + 1205741205742120592120592F210076651007665119879119879 (9)



75 100 200 300 400 500

Temperature (K)

50

100

1000

10000

(MW

mminus

2K

minus1)

DMMAlCu

AlCu

PdIr

NiZr

NiTi

TiNMgO

1014 1015 1016 101710

1000

10000

AlCu-785 K

AlCu-298 K

(MW

mminus

2K

minus1)












ℎK12 = 10076501007650ℎminus1K1int + ℎ

minus1Kint210076661007666

minus1 (10)

0 2 4 6 8 10 12 14 16 18 20


100

1000

5000

AlCu

AlCu

(as received)

CrSi

VCDMM

approximation

to DMM

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )







(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















12057712057712 = 11990311990321 = 1 minus 12057712057721 (5)


1199021199021 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895119891119891119891119891111989511989512057712057712119889119889120598120598 (6)


ℎK12 =121005576100557611989511989510045601004560120598120598max119895119895

120598120598min1198951198951205981205981198951198951198631198631119895119895

120597120597119891119891120597120597120597120597

119891119891111989511989512057712057712119889119889120598120598 (7)


12057712057712 =11990211990221

11990211990212 + 11990211990221 (8)







ℎK =1205741205741120592120592F11205741205742120592120592F2

4 100764910076491205741205741120592120592F1 + 1205741205742120592120592F210076651007665119879119879 (9)



75 100 200 300 400 500

Temperature (K)

50

100

1000

10000

(MW

mminus

2K

minus1)

DMMAlCu

AlCu

PdIr

NiZr

NiTi

TiNMgO

1014 1015 1016 101710

1000

10000

AlCu-785 K

AlCu-298 K

(MW

mminus

2K

minus1)












ℎK12 = 10076501007650ℎminus1K1int + ℎ

minus1Kint210076661007666

minus1 (10)

0 2 4 6 8 10 12 14 16 18 20


100

1000

5000

AlCu

AlCu

(as received)

CrSi

VCDMM

approximation

to DMM

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )







(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














ℎK =1205741205741120592120592F11205741205742120592120592F2

4 100764910076491205741205741120592120592F1 + 1205741205742120592120592F210076651007665119879119879 (9)



75 100 200 300 400 500

Temperature (K)

50

100

1000

10000

(MW

mminus

2K

minus1)

DMMAlCu

AlCu

PdIr

NiZr

NiTi

TiNMgO

1014 1015 1016 101710

1000

10000

AlCu-785 K

AlCu-298 K

(MW

mminus

2K

minus1)












ℎK12 = 10076501007650ℎminus1K1int + ℎ

minus1Kint210076661007666

minus1 (10)

0 2 4 6 8 10 12 14 16 18 20


100

1000

5000

AlCu

AlCu

(as received)

CrSi

VCDMM

approximation

to DMM

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )







(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















ℎK12 = 10076501007650ℎminus1K1int + ℎ

minus1Kint210076661007666

minus1 (10)

0 2 4 6 8 10 12 14 16 18 20


100

1000

5000

AlCu

AlCu

(as received)

CrSi

VCDMM

approximation

to DMM

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )







(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















50 100 200 300 400 50050

100

200

300

400

500

Temperature (K)

= 114 nm

= 65 nm

= 06 nm

lt 01 nm

No oxide

[89]

DMM

DMM+ oxide

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1 )







(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(11)



minus1 (12)





005 01 1 10 20

RMS roughness (nm)

0

25

50

75

100

125

150

175

200

225


Chemical etch

[21]

Quantum dots

[19]

Chemical etch[20]

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)










75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














75 100 200 300 400 500

Temperature (K)

1

10

100

1000

DMM

GaSbGaAs

TiNMgO

GaSbGaAs

Bidiamond

GaSbGaAs

Ther

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)














50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















50 100 200 300 400 500

Temperature (K)

10

20

30

40

50

60

70

80

90

100

DMM-2D variation

DMM-2D variation

with bonding

AlO-diamond

AlO-SLG

AlSLG

AldiamondTher

mal

bo

un

dar

y co

nd

uct

ance

(M

W m

minus2

Kminus

1)



119902119902SLG =121198861198861005576100557611989511989510045601004560120596120596119895119895119895119895119895119895

0ℏ1205961205961205961205962D119895SLG119891119891119891119891SLG119889119889120596120596119895 (13)






10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











10 Summary



Acknowledgments


References



































































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









3FWJFX SUJDMF …

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