Copper-graphene Composite Heat Sink

Thermal Conductivity of Copper-Graphene Composite FilmsSynthesized by Electrochemical Deposition with ExfoliatedGraphene Platelets

K. JAGANNADHAM

Samples of graphene composites with matrix of copper were prepared by electrochemicalcodeposition from CuSO4 solution with graphene oxide suspension. The thermal conductivity ofthe composite samples with different thickness and that of electrodeposited copper was deter-mined by the three-omega method. Copper-graphene composite films with thickness greaterthan 200 lm showed an improvement in thermal conductivity over that of electrolytic copperfrom 380 W/m.K to 460 W/m.K at 300 K (27 �C). The thermal conductivity of copper-graphene films decreased from 510 W/m.K at 250 K (–23 �C) to 440 W/m.K at 350 K (77 �C).Effective medium approximation (EMA) was used to model the thermal conductivity of thecomposite samples and determine the interfacial thermal conductance between copper andgraphene. The values of interface thermal conductance greater than 1.2 GW/m2.K obtainedfrom the acoustic and the diffuse mismatch models and from the EMA modeling of theexperimental results indicate that the interface thermal resistance is not a limiting factor toimprove the thermal conductivity of the copper-graphene composites.

DOI: 10.1007/s11663-011-9597-z� The Minerals, Metals & Materials Society and ASM International 2011

I. INTRODUCTION

LOW-COST manufacturing of thermal interfacematerials and heat spreaders with high capacity todissipate thermal energy is of significant importance innanoelectronics and high-frequency and high-powerdevices such as power amplifiers and laser diodes.[1–3]

The thermal conductivity of diamond,[4] graphite,[5]

carbon nanotubes (CNTs),[6] and graphene[7,8] is highercompared with other materials. Of these, the measure-ments at room temperature showed that the thermalconductivity of free-standing graphene is the highestwith the value between 3000 and 5000 W/m.K and, thus,greater than that of other forms of carbon discovereduntil now. However, it was also shown recently[9] thatthe thermal conductivity of monolayer of graphenesupported on SiO2 is only 600 W/m.K as a result ofphonon leaking and phonon scattering by the substrate.

To alleviate the problems associated with low thermalconductivity thermal interface materials[10] (TIM),recently we studied a simple method of processingIn-graphene and In-Ga-graphene composites[11,12] andfound that the thermal conductivity at 300 K (27 �C) isimproved by a factor of 2.5 and 3, respectively,compared with that of In and In-Ga. In general, a heatspreader system contains a high capacity heat spreaderand an external cooling system in addition to the TIMattached to the active device region. In the current work,

we studied the preparation of copper-graphene (Cu-gr)composite heat spreaders by electrochemical codeposi-tion from CuSO4 solution with graphene oxide suspen-sion. The thermal conductivity of the Cu-gr compositeswas determined experimentally and modeled to deter-mine the importance of interface thermal conductanceand graphene particle size.

II. EXPERIMENTAL PROCEDURES

The preparation of copper-graphene composite sam-ples has been described at length in the previouswork.[11,13] A brief description is provided here. Graph-ene oxide (GO) films were prepared by chemicalexfoliation from microcrystalline graphite[14] suppliedby Asbury Graphite Inc. (Asbury, NJ) The details ofpreparation of exfoliated GO are discussed at length inour previous work.[11] A suspension of GO particulatesin a 0.2 M solution of CuSO4 was prepared. Electro-chemical codeposition from a bath containing grapheneoxide (GO) suspension in a solution of technical gradeCuSO4 in distilled water was carried out on oxygen-freehigh-conductivity (OFHC) copper foils. The pH wasmaintained close to 7, and a low current density of1.75 mA/cm2 and growth rate of 2 to 3 lm/h[13] wereused with a pure Cu anode to achieve smooth films. GOis hydrophobic on the basal plane and hydrophilic at theedges so that the suspension of the GO particulates ismaintained at pH> 6.[15] Five samples of Cu-graphene(Cu-gr) composite with smooth surfaces were deposited,each with slightly different thickness and were cut tosmaller size of approximately 8-mm 9 10-mm areausing a diamond saw. To evaluate the improvement in

K. JAGANNADHAM,Associate Professor, is with the Departmentof Materials Science and Engineering, North Carolina State University,Raleigh, NC 27695. Contact e-mail: [email protected]

Manuscript submitted August 14, 2011.Article published online November 10, 2011.

316—VOLUME 43B, APRIL 2012 METALLURGICAL AND MATERIALS TRANSACTIONS B

the thermal conductivity introduced by graphene, sam-ples of electrolytic Cu were also deposited on the Cu foilfrom the same electrolytic bath but without GOsuspension. The thickness of OFHC Cu foil in all thesamples was 135 lm, but the thickness of the depositedcomposite film on the opposite sides of the Cu foilvaried.

The electrolytic Cu and Cu-gr samples were heated inflowing hydrogen atmosphere at partial pressure of20 Torr and 673 K (400 �C) for 3 hours to reduce GOto graphene and Cu2O or CuO to Cu and thus formCu-gr composites. Additional characterization by scanningelectron microscopy (SEM) for graphene morphologyand distribution of graphene platelets, energy dispersivespectrometry (EDS) for presence of oxygen and otherimpurities, and electrical conductivity as a function oftemperature were carried out.[13] The electrical resistivityand temperature coefficient of resistance (TCR) wereused previously to determine the volume fraction andresistivity of thinner films of graphene in all thesamples.[13] SEM imaging combined with lineal fractionanalysis in quantitative metallography[16] was used todetermine the volume fraction of thicker graphene in theCu-gr samples.

Thermal conductivity of the samples was measuredusing the 3-x method. The details of the 3-x method aredescribed at length in the previous work[12] and in theoriginal Reference 17. A schematic diagram illustratingthe sample with a gold heater line on the surface ispresented in Figure 1. In the 3-x method, a line heater isused both as a source of heat and to measure thetemperature on the surface of the sample for which thethermal conductivity is measured.[12,17] The Cu-gr com-posite sample surface is isolated by spreading anelectrically insulating polymer film. The polymer filmis a commercial thermosetting white epoxy S-30/3045supplied by Devcon (Ellsworth Adhesives, Germantown,WI). In addition, thin films of insulating Si and ZrO2

each with thickness approximately 0.2 lm were depos-ited by laser physical vapor deposition (LPVD) on thetop of the polymer film to enable the adhesion of a goldheater line to the surface. The gold heater line, with awidth below 200 lm and length up to 8 mm, wasdeposited by LPVD using a negative stainless steel maskplaced on the surface. Indium foils, with a width of1 mm and length of 5 mm, were pressed on to either endof the gold heater line to make electrical contacts using

gold wires of 0.1 mm diameter. Two contacts on theopposite sides of the heater line were used to input thecurrent and the other two were used to measurethe output of voltage.The samples were mounted using ZnO paste on the

substrate stage of a temperature-controlled vacuumchamber supplied by MMR Technology (MountainView, CA). The chamber was kept under vacuum at6 millitorr using a mechanical pump during electricalmeasurements. The temperature of the sample stage wascontrolled to ± 0.1 K by K-20 controller in conjunctionwith high-pressure nitrogen gas that was passed througha nozzle in the stage system of the chamber or by theJoule-Thomson effect. The measurements were made atthree values of temperature, 250 K, 300 K, and 350 K(–23 �C, 27 �C, and 77 �C) on two samples and at 300 K(27 �C) on all the samples. After the set temperature wasreached, the internal source of the lock in amplifier wasused to supply power at the set input voltage V1 andfrequency f, and the output voltage V3 was measured atfrequency 3f. SR 830 DSP lock in amplifier supplied byStanford Research Systems (Sunnyvale, CA) was usedto supply the power at frequency f and measure thevoltage at frequency 3f.The measurements were made as a function of fre-

quency f (2pf = x) that was increased from 1 Hz to3.5 kHz in different steps to cover the complete range.The electrical resistance (R) and temperature coefficientof resistance (a = dR/RdT) of the heater line weremeasured from the four contacts, with the two outercontacts to supply current at a constant value and theother two to measure the voltage. Keithley source meter2400-LVand the nanovoltmeter 2182 (Keithley, Cleveland,OH) were used with reverse current procedure to cancelout the thermal electromotive force at the contacts. Theaverage of the absolute value of the voltage was used todetermine the resistance using Ohm’s law. The temper-ature was varied from 290 K to 315 K (17 �C to 42 �C) insteps of 5 K to determine the value of a.The increment in temperature of the heater line as a

function of frequency was calculated using[17]

dT ¼ 2V3= aV1ð Þ ½1�

where V1 is the voltage at frequency f, V3 at frequency3f, and a is the temperature coefficient of resistance ofthe heater line. The power input per unit length into theheater was determined from V1

2/(lR) where l is thelength of the heater line. A graph of dT per unit powerinput was generated as a function of ln(f) over thecomplete range of frequency at a given substratetemperature. These results were curve fitted using amultilayer analysis[18] with the help of a computerprogram. The parameters in the multilayer analysisconsist of the number of layers, thickness of each layer,thermal conductivity and heat capacity of each layer,and the half width of the gold heater line. Interfacialthermal resistance between layers was also included inthe multilayer analysis by introducing a layer with smallthickness, heat capacity, and finite thermal conductivity.The multilayer analysis was started using the known

values of thermal conductivity and heat capacity of

Fig. 1—Schematic illustration of the sample for the 3-x measure-ments. The sample with electrically insulating films on the top andthe gold heater with indium foils attached on either side are shown.The gold wire electrical contacts on either side of the heater are alsoshown.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 43B, APRIL 2012—317

different layers and the thickness of different layersshown schematically in Figure 2. The values of theparameters were modified during the numerical analysisto fit the experimental data and thus determine thethermal conductivity of Cu-gr layers. The polymer filmhad low thermal conductivity. The thermal wavespenetrated into the ZnO paste only when the polymerfilm was thin. Therefore, an additional set of measure-ments were also made on two samples with thinnerinsulating polymer film. The thicknesses of the polymerfilm and the insulating layer are given in the results ofcurve fitting described subsequently.

III. EXPERIMENTAL RESULTS

A. Results of Characterization of the Cu-grComposite Films

An SEM image of the cross section of the compositesample is provided in Figure 3 along with the energydispersive spectrometry of the graphene film in Cu inFigure 4. The SEM image in backscattering mode andthe X-ray maps of the carbon and the Cu signals areshown in Figure 3. The labeling in Figure 3 shows thepure Cu foil in the center and the Cu-gr composite filmsoutside. The image also contains some graphene partic-ulates that were sticking on the surface of the pure Cufilm in the center as a result of contamination fromcutting with diamond saw. It is observed from thisimage that graphene is distributed uniformly in thecomposite films outside the Cu foil. The X-ray map ofcarbon signal is shown in the middle of Figure 3.

The carbon signal in the center region of Cu is to theresult of instrumental noise. If the noise from the signalis taken into account, then the middle region of Cu hasno carbon but the outer regions of Cu-gr have a highercarbon signal. The Cu signal shown on the right-handside of Figure 3 contains strong Cu signal from thecenter and a reduced Cu signal from Cu-gr composite.The EDS spectrum obtained from graphene particu-

lates in the Cu-gr film is shown in Figure 4. Thespectrum contains strong C and Cu peaks and a small Opeak. The oxygen peak is associated with residualconcentration in the film. X-ray diffraction[13] from alarge area of the Cu-gr films in the as-depositedcondition showed a diffraction peak at 2h = 11.8 degthat is associated with graphene oxide particulates butdid not show any peak associated with graphite at2h = 26.5 deg. This result indicated that the compositesamples did not contain any graphite. However, the GOpeak was not present on annealing in hydrogen at 673 K(400 �C) for 3 hours, which indicated that GO isconverted to graphene layers of small thickness. Trans-mision electron microscopy[13] of the suspended GOparticulates on microscopy grid also showed thin filmswith a diffraction pattern containing diffraction spotsarranged in a hexagonal pattern. These results presentedin the previous work[13] and the SEM characterization

Fig. 2—Schematic illustration of the different layers in the cross sec-tion of the composite samples prepared for thermal conductivitymeasurement. The gold heater wire and indium contacts are notshown.

Fig. 3—SEM image of the cross section of composite sample of Cu-gr layers grown on Cu foil with the three regions labeled is shown on theleft. The image was taken at a magnification of 100 times. The map of carbon signal is shown in the middle. The map of Cu signal is shown onthe right. The carbon signal in the Cu foil region in the middle is caused by the electronic noise in the instrument. The graphene particles in theCu foil region in the center were present because of contamination during cutting with the diamond wheel.

Fig. 4—EDS spectrum of graphene particulates on Cu matrix inCu-gr composite films. The residual oxygen peak is observed in addi-tion to the major carbon and Cu peaks.


confirmed that the composite films contained grapheneplatelets in a Cu matrix.

B. Results of Thermal Conductivity

A list of all the samples and the volume fractions ofgraphene evaluated from the measurement of electricalresistivity[13] and quantitative metallography are pro-vided in Table I. First, the results obtained on thethermal conductivity of two samples of Cu-gr4 using athinner and a thicker polymer insulating film on the topare described. Next, the results obtained on all thesamples are presented along with the temperaturedependence.

The results obtained at 300 K (27 �C) from sampleCu-gr4-1, which consisted of Cu-gr composite films ofthickness 240 lm deposited on the top and 205 lm atthe bottom of the Cu foil are shown in Figure 5. Thethermal conductivity, heat capacity, and thickness of thedifferent layers used in the multilayer analysis are listedin Table II. The heat capacity of Cu and ZnO are takenfrom Reference 19. The thickness of the polymer filmwas 27.5 lm and that of the insulating ZrO2, Si, and thepolymer was 11.8 lm. Thus, the total thickness of thetop insulating films was 39.3 lm. An interfacial layer of0.01 lm was used to represent the thermal resistancebetween the gold heater line and the top of ZrO2 film.The thermal waves penetrated to the bottom ZnO layerused for mounting the sample on the substrate holder inthe chamber. The region of the graph with smaller slopein Figure 5 with 0< ln(f)< 5 represents the bottomlayers consisting of Cu-gr, Cu, Cu-gr, and ZnO. Theregion with the higher slope with 5<ln(f)< 8.2 repre-sents the top layers of electrically insulating films.

The results obtained at 300 K (27 �C) from sampleCu-gr4-2, which consisted of Cu-gr composite film ofthickness 215 lm deposited on the top and 165 lm atthe bottom of the Cu foil, are shown in Figure 6. Thethermal conductivity, heat capacity and thickness of thedifferent layers used in the multilayer analysis are listedin Table III. The thickness of the polymer film was66.1 lm and that of the insulating ZrO2, Si, andpolymer layer was 13.2 lm. Thus, the total thickness

of the top insulating films was 79.3 lm, which is largerthan that in the sample Cu-gr4-1. As stated, aninterfacial layer of thickness 0.01 lm was used torepresent the thermal resistance between the gold heaterline and the ZrO2 film. The thermal waves have not

Table I. List of Samples Tested with Values of the Volume Fraction of Graphene fg Evaluatedfrom Electrical Resistivity and Metallography*

Sample

fg

Total Thickness(lm)

Thermal Conductivity (102 W/m.K)

Resistivity MetallographyT = 250 K(–23 �C)

T = 300 K(27 �C)

T = 350 K(77 �C)

Cu foil 0 0 135 4.2 3.9 3.8Cu-elec 0 0 425 4.0 3.8 3.7Cu-gr3 0.08 0.25 250 — 4.5 —Cu-gr4 0.09 0.24 445 5.1 4.6 4.4Cu-gr5 0.11 0.26 395 — 4.6 —Cu-gr6 0.11 0.19 385 4.6 —Cu-gr7 0.09 0.20 432 — 4.6 —

*The values of thermal conductivity of OFHC Cu foil, electrolytic Cu (Cu-elec), and Cu-gr composite samples determined by the 3-x method andanalyzed by multilayer analysis are shown. The values of thermal conductivity are accurate to ±0.1. Cu-elec is electrolytic Cu Deposited on OFHCCu foil of thickness 135 lm.

Fig. 5—Blue diamond symbols represent the experimental values ofdT per unit power input and red squares represent the resultsobtained from multilayer analysis for sample Cu-gr4-1 with thinnerpolymer film. Temperature of the substrate was 300 K (27 �C).Table II shows the different parameters used in the multilayer analysis.

Table II. Values of the Different Parameters Used to Fit theExperimental Results Obtained in the Sample Cu-gr4-1 at

300 K (27 �C) with the Heater Line Half Width of 65 lm and

Length of 8 mm*

Identified Layer

ThermalConductivity(102 W/m.K)

HeatCapacity

(106 J/m3.K)Thickness

(lm)

Interface layer 0.0010 0.001 0.01ZrO2, Si and polymer 0.0060 2.0 11.8Polymer layer 0.0503 2.0 27.5Cu-gr Composite 4.6 3.5 240Cu foil 3.9 3.5 135Cu-gr Composite 4.6 3.5 205ZnO 1.0 2.96 1200

*The results of curve fitting are shown in Fig. 5. The polymer filmplus the insulating layer on the top is relatively thin (39.3 lm). Thetemperature of the substrate was 300 K (27 �C).


penetrated into the ZnO layer used for mounting thesample on the substrate holder because the polymerlayer with lower thermal conductivity has lower thermaldiffusion depth. The region of the graph with smallerslope in Figure 6 with 0< ln(f)< 1 represents thebottom layers consisting of Cu-gr, Cu, and Cu-gr. Theregion with higher slope with 1< ln(f)< 8.2 representsthe top insulating layers.

The multilayer analysis of the Cu-gr4-1 and Cu-gr4-2samples with a different thickness of polymer film wascarried out so that the larger region of smaller slope, asshown in Figure 5, could be curve fitted with moreconfidence. However, the values of the thermal conduc-tivity of the Cu-gr layers shown in Tables II and III aresame indicating that the analysis in Figure 6 is equallyvalid. The thickness of the Cu-gr composite film in boththese samples is large (>200 lm), so that the thicknessdependence of thermal conductivity did not change thevalue of thermal conductivity obtained in these twosamples significantly.

A multilayer analysis was also performed on the 3-xexperimental data obtained from two samples of elec-trolytic copper (Cu-elec) film deposited on the Cu foil butwith thinner and thicker electrically insulating polymer

film provided on the surface. Different parameters usedin the analysis and the curve-fitted graphs are notpresented in this article for the sake of brevity. Theresults were found to be the same from both samples, andthe value of thermal conductivity is shown in Table I.A multilayer analysis of 3-x experimental data

obtained at 300 K (27 �C) from samples of Cu-gr3,Cu-gr5, Cu-gr6, and Cu-gr7 was performed, and theresults are shown in Table I. The thermal conductivityof Cu-gr film in sample Cu-gr3 was found to be slightlylower than that of other samples, and this result isattributed to slightly variable microporosity introducedduring electrochemical deposition.The temperature dependence of the thermal conduc-

tivity of the samples Cu-gr4-1 and Cu-elec with thinnerpolymer-insulating film on the surface was determinedby collecting the experimental data at 250 K (–23 �C)and 350 K (77 �C) in the 3-x setup described previously.The results were curve fitted using the multilayeranalysis. The values of the different parameters used inthe multilayer analysis and the curve fitted graphs arenot presented for the sake of brevity. The values of thethermal conductivity obtained for the Cu-gr films areshown in Table I.

IV. DISCUSSION

The results presented in Table I show that the thermalconductivity of electrolytic Cu is 3.8 W/cm.K at 300 K(27 �C) compared with the value of 3.9 W/cm.K for Cufoil on which it is deposited. The slightly lower value isassociated with a higher electrical resistivity[13] that isalso observed in the electrodeposited Cu. An EDSanalysis[13] of the samples showed presence of higheroxygen level in all the electrodeposited samples, and it isattributed to the impurities in the technical grade CuSO4

used in the electrochemical deposition. The details of theimpurities and the higher electrical resistivity observedin the electrodeposited films are presented in theprevious work.[13] Therefore, a subsequent comparisonof thermal conductivity of Cu-gr films will be made withrespect to the electrodeposited Cu film instead of theOFHC Cu foil.The thermal conductivity of all the films of Cu-gr is

found to be 4.6 W/cm.K except for a slightly smallervalue of 4.5 W/cm.K in the sample Cu-gr3 with smallertotal thickness of the composite film, as shown inTable I. This result indicates a large improvement from3.8 W/cm.K associated with electrolytic Cu. The tem-perature dependence of thermal conductivity of theCu-gr film in the sample Cu-gr4-1 is similar to that ofelectrolytic Cu in the temperature range of 250 K to350 K (–23 �C to 77 �C). It is lower at higher temper-ature[20,21] as a result of increased scattering of electronsin Cu and phonons in graphene. These results areanalyzed using effective mean field approximation(EMA). In particular, EMA is used to determine thedependence of thermal conductivity on the volumefraction of graphene incorporated in the Cu-gr compos-ite films and to evaluate the interface thermal conduc-tance between Cu and graphene.

Fig. 6—The blue diamond symbols represent the experimental valuesof dT per unit power input. The red circles represent the resultsobtained from multilayer analysis for sample Cu-gr4-2 with thickerpolymer film. The temperature of the substrate was 300 K (27 �C).Table III shows the different parameters used in the multilayeranalysis.

Table III. Values of the Different Parameters used to Fit theExperimental Results Obtained in the Sample Cu-gr4-2 at

300 K (27 �C) with the Heater Line Half Width of 94 lm and

Length of 8 mm*

Identified Layer

ThermalConductivity(102 W/m.K)

HeatCapacity

(106 J/m3.K)Thickness

(lm)

Interface layer 0.0010 0.001 0.01ZrO2, Si and Polymer 0.00295 2.0 13.2Polymer layer 0.0070 2.0 66.1Cu-Gr Composite 4.6 3.5 215Cu foil 3.9 3.5 135Cu-Gr Composite 4.6 3.5 165

*The results of curve fitting are shown in Fig. 6. The polymer filmplus the insulating layer on the top is relatively thick (79.3 lm). Thetemperature of the substrate was 300 K (27 �C).


A. Evaluation of Volume Fraction of Graphene

The volume fraction of graphene determined from alineal fraction analysis using quantitative metallogra-phy[16] is presented in Table I. SEM imaging of graph-ene in backscattering mode showed only those grapheneplatelets that are thicker exhibit dark contrast. As aresult, thinner graphene platelets such as single orbilayer graphene that exhibit weak contrast cannot beidentified. Therefore, the volume fraction estimatedfrom quantitative metallography is slightly lower thanthe actual value because the graphene films of smallerthickness are not included.

The volume fraction of graphene determined fromelectrical resistivity and temperature coefficient of elec-trical resistance (TCR)[13] is also presented in Table I.The two unknowns, resistivity and the volume fractionof graphene, were solved using resistivity and the TCRof the composite samples in terms of the formulations ofEMA.[13] Although this method is helpful to evaluatethe volume fraction of thinner graphene platelets thatcontribute significantly to electrical conductivity, thickergraphene platelets that do not contribute to improve-ment cannot be evaluated. The electrical conductivity ofgraphene platelets thicker than 5 to 10 atomic layersapproaches the characteristics of graphite that is semi-metallic with lower electrical conductivity. Therefore,thicker graphene platelets do not contribute to electricalconductivity. However, both thinner and thicker graph-ene platelets contribute significantly to thermal conduc-tivity[22–24] because thermal conductivity in the basalplane of graphene platelet is much higher than that ofCu. Therefore, the volume fraction of graphene thatshould be included in the analysis of thermal conduc-tivity of Cu-gr composites is higher than the valueobtained from quantitative metallography. The volumefraction of thinner and thicker graphene platelets in thesample is determined by adding the values obtainedfrom electrical resistivity and quantitative metallogra-phy, respectively. The volume fraction of graphenedetermined by electrical conductivity is much smallerand within the accuracy achieved by quantitativemetallography, and therefore, the total volume fractionis a good measure of the graphene present in the sample.The results are expected to be accurate within ±0.05.

B. Effective Medium Approximation

EMA has been used previously to explain the thermalconductivity of indium-graphene composites and carbonnanotube composites.[12,25] Therefore, the formulationswill be described only briefly. Under this approximation,the isotropic thermal conductivity of the Cu-gr com-posite medium is given by

Kc ¼ Kg Km 3� 2fg� �

þ 2fgKg

� �

= fgKm þ Kg 3� fg þ afg� ��

½2�

where fg is the volume fraction of graphene and K standsfor thermal conductivity, with subscript c for thecomposite (Cu-gr), g for graphene, and m for matrixor Cu. The parameter a is given by Km/hr where h is theinterfacial thermal conductance between the matrix and

graphene and r is the effective radius of the grapheneplatelets. In the current analysis, r is taken as half theaverage size of graphene platelets present in the com-posite samples. EMA is used to determine the interfacialthermal conductance, h from Eq. [2].The experimental determinations[8,22] of Kg have

shown that its value is high when the size of thegraphene platelets is bigger, such as 5 lm. A modelinganalysis[21,23] also indicates that the value is expected tobe in the experimentally observed range. Modeling[24]

showed also that the value of Kg decreases to 1000 W/m.Kwhen the size of the graphene platelet is 1 lm. The size(L) dependence of thermal conductivity in graphenenanoribbons has been shown[25] to follow (L)b, where bremains between 0.35 and 0.45. In addition, Klemenspointed out[26,27] that thermal energy leaks in to thematrix when graphene is present on a substrate, thereby,the thermal conductivity will be reduced further by 20 to50 pct if the phonon velocity in the matrix is smallerthan that in graphene. This result has been confirmedrecently from the thermal conductivity measurements ofgraphene suspended on SiO2.

[9]

The average size Lav of the graphene platelets and theweighted average of the thermal conductivity Kg weredetermined from the distribution of the platelet sizes inthe graphene suspension.[12] The same graphene suspen-sion is used in the current experiments so that theaverage size of graphene platelets is 0.64 lm and theaverage value of Kg is 800 W/m.K. EMA is performedwith these values of the parameters and in particulara = Km/hr where r = 0.32 lm. The results of interfa-cial thermal conductance h obtained from Eq. [2] areshown in Table IV.The results shown in Table IV illustrate that the

interface thermal conductance remains between 1.2 and2.2 in units of 109 W/m2.K, which is reasonably high.The variation in h between the samples is caused by thedifficulty to determine the volume fraction of grapheneaccurately in the composites.

C. Interface Thermal Conductance from Acousticand Diffuse Mismatch Models

The thermal conductivity in Cu is electron mediated,whereas that in graphene is phonon mediated in thetemperature range of 250 K to 350 K (–23 �C to 77 �C).

Table IV. Value of Interfacial Thermal Conductance hDetermined from Eq. [2] in all the Cu-gr Samples using the

Values of Kc and fg from Table I, Km = 380 W/m.K,

Kg = 800 W/m.K, and r = 0.3 lm*

CompositeSample

Kc

(102 W/m.K) fg

h(109 W/m2.K)

Cu-gr3 4.5 0.33 1.2Cu-gr4 4.6 0.33 1.5Cu-gr5 4.6 0.37 1.3Cu-gr6 4.6 0.30 1.9Cu-gr7 4.6 0.29 2.2

*fg in this table is the sum of the two values of the volume fractionshown in Table I. See the explanation in Section IV.


The interface thermal conductance is evaluated previ-ously[12] using two approaches. In the first, we considerelectron–phonon coupling in Cu and phonon–phononcoupling between Cu and graphene. Next, electron–electron coupling between Cu and graphene andelectron–phonon coupling in graphene are used in thesecond approach.

The electron–phonon coupling term hep in Cu isdetermined using[28]

Gep ¼ Cem=s and hep ¼ GepKpm

� �1=2 ½3�

where Cem is the electronic specific heat of Cu and s isthe relaxation time for electron–phonon energy transfer.The values of different parameters used in themodeling areprovided in Table V.[29–32] The electronic specific heat iscalculated using the Cem = cT where c = 97.0 J/m3.Kfrom Table V.[29] The relaxation time is assumed to be1 ps[28] so that Gep = 29.1 9 106 GW/m3 at 300 K(27 �C). The phonon thermal conductivity of Cu iscalculated from Kpm = Clmvdmlp/3 where Clm is thelattice specific heat, vdm is the Debye phonon velocity,and lp is the phonon mean free path taken to be 5 nm.[28]

Using the values listed in Table V, Kpm = 16.2 W/m.K.The contribution to the interfacial thermal conductancefrom electron–phonon coupling hep is 6.87 9 108 W/m2.K.

The phonon–phonon contribution hpp to the interfacethermal conductance is determined from the acoustic orthe diffuse mismatch models.[33,34] In the acoustic mis-match model[33]

hpp ¼ Clmvdma=4 ½4�

where a is the transmission coefficient determined from

a ¼ 4ZmZg= Zm þ Zg

� �2 ½5�

where Z is the acoustic impedance of each medium givenby the product of density and Debye velocity, and thesubscripts m and g refer to Cu and graphene, respec-tively. The value of hpp from acoustic mismatch model isdetermined using the values of parameters listed inTable V and found to be 2.26 9 109 W/m2.K. Theexpression for hpp in the diffuse mismatch model[33] is

hpp ¼ 1:02 � 1010T3 R 1=v2m� �

R ð1=v2gn o

= R ð1=v2m þ 1=v2g

n o½6�

In Eq. [6], the summation is performed over thelongitudinal and the two transverse modes in Cu but

only over the longitudinal and the single transversemode in graphene because it is a two-dimensionalmedium. Using the parameters given in Table V, thevalue of hpp is found to be 1.50 9 109 W/m2.K. The netthermal conductance[28] determined from 1/h = 1/hep+1/hpp is 4.7 9 108 W/m2.K from the diffuse mismatchmodel (DMM) and 5.3 9 108 W/m2.K from the acous-tic mismatch model (AMM).An alternative expression from diffuse mismatch

model[34] is,

hpp ¼ ClmvdmClgvdg=4 Clmvdm þ Clgvdg� �

½7�

where C represents, as before, the lattice specific heatand vd the Debye velocity. Using the parametersprovided in Table V, the value of hpp = 18.9 9108 W/m2.K. In this case, the total thermal conductanceis found to be 5.0 9 108 W/m2.K. From the precedingresults, the values of h from the acoustic and the diffusemismatch models are close.In the next representation, the total thermal conduc-

tance will consist of electron–electron coupling betweenCu and graphene and electro–phonon coupling ingraphene. The first part is determined from,[35]

hee ¼ CegvfgCemvfmT=4 Cegvfg þ Cemvfm� �

½8�

where vfg represents the Fermi velocity in graphene andvfm in Cu, Ceg is the electronic specific heat of grapheme,and Cem is that of Cu. The electronic specific heat isdetermined as before from Ce = cT and the values of care listed in Table V for each medium so that the value ofhee atT = 300 K(27 �C) is foundtobe18.6 9 1010 W/m2.K.The electron–phonon coupling term[28] in graphene hep isdetermined as before from hep = {Gep Kpg}

1/2 whereGep = (Ceg/s) and Kpg is the phonon thermal conductiv-ity in graphene. Using the parameters listed in Table V,Gep = 2.38 9 106 GW/m3.K. The phonon thermal con-ductivity Kpg in graphene is size dependent. The distribu-tion of graphene platelet sizes in the suspension isevaluated, and using the size dependence[25] of thermalconductivity, the average value was shown previously[12]

to be 800 W/m.K so that hep = 1.38 9 109 W/m2.K. Thenet thermal conductance h given by 1/h = 1/hee+1/hep isfound to be 1.37 9 109 W/m2.K. Therefore, the value ofthermal conductance basedon electron–electron couplingis higher than when phonon–phonon coupling is usedbetween Cu and graphene.The different values obtained for the interface thermal

conductance from the various components and the total

Table V. Parameters used in the Calculation of the Interface Thermal Conductance*

MediumDensity(106 gm/m3)

Velocity (103 m/s)

Lattice SpecificHeat (106 J/m3.K) c (103 J/m3.K2) vf (10

6 m/s)Longitudinal

vl

Transversevt

Debyevd

Cu 8.93 4.8 2.4 2.8 3.5 0.097 1.57Graphene 2.26 24.0 16.0 18.6 1.84 0.008 1.10

*Cu is treated as three-dimensional medium (3/vd2 = 1/vl

2+2/vt2) and graphene as two-dimensional medium (2/vd

2 = 1/vl2+1/vt

2) for calculatingthe Debye velocity vd. The values of acoustic velocity vl and vt are taken from Refs. 29 and 30 for Cu and from Refs. 19 and 27 for graphene. Thevalues of c and the Fermi velocity vf are taken from Refs. 29 and 30 for Cu and from Refs. 31 and 32 for graphene.


thermal conductance using the two approaches of carriercoupling are listed in Table VI. Although the electron–phonon coupling term in graphene is weak, it is strongerthan in Cu by a factor of two. Similarly, the DMMindicates that electron–electron coupling term is strongerthan phonon–phonon coupling term between Cu andgraphene. The value of the interface thermal conduc-tance, from Table VI, is lower than the experimentalvalue shown in Table IV when phonon–phonon couplingis considered between Cu and graphene. The valueobtained from electron–electron coupling, as shown inTable VI, is closer to the experimentally determined valueusing EMA. The lower Debye velocity of sound in Cucomparedwith that in graphene by almost a factor of 7, asshown in Table V, indicates clearly that quenching ofphonons in graphene will take place.[27] Therefore, thethermal conductivity Kg of single- and double-layergraphene platelets will be reduced both from quenchingof phonons and smaller size of graphene platelets. Fromthis point of view, graphene platelets of 4 to 10 atomiclayers thick wherein the phonon transport within innerlayers is still effective seem to be advantageous. It isimportant to determine the thickness dependence ofthermal conductivity of graphene platelets, especiallywhen in contacts with a matrix such as Cu.

It is useful to compare the current results with that ofinterface thermal conductance between metals andgraphite obtained experimentally[36,37] and using mod-eling.[38] Although, these studies did not examinedirectly the interface thermal conductance between Cuand graphite, the results obtained for Al, Au, Ti, and Cron C-axis-oriented graphite are relevant. The resultsshowed that the interface thermal conductance remainedbetween 30 and 100 MW/m2.K with lower range valuesapplicable to noninteracting face-centered cubic metalslike Au and higher values for interacting metals like Tiand Cr. The larger value of 1.0 9 108 W/m2.K in thesemeasurements and modeling using phonon cou-pling[36,37] is only smaller by a factor of 5 from theDMM approach used in the current work. The maindiscrepancy is associated with the C-axis orientation ofgraphite crystals on which these studies were performed.The thermal conductivity of a few layers of graphene isthree orders of magnitude smaller in the C-directioncompared with the value in the basal plane or normal tothe C-direction. The values shown for the present workassume thermal transport in the basal plane of grapheneplatelets, and hence, the interface thermal conductanceis also expected to be much higher.

The modified DMM approach with phonon couplingused in Reference 38 estimates the interface thermalconductance between basal plane graphite and Al at300 K (27 �C) to be close to 4 9 108 W/m2.K. Thisvalue is close to that obtained in the current calculationsusing phonon–phonon coupling in the DMM approach.The thermal conductivity of Cu is a factor of 2 higherthan that of Al and that of graphene is also higher thanthat of graphite so that the interface thermal conduc-tance between Cu and graphene is expected to be largerthan that between Al and graphite. More importantly,the thermal conductance using electron–electron cou-pling seems to be higher than that using phonon–phonon coupling between Cu and graphene. The higherexperimental value of interface thermal conductance inthe current analysis using EMA could arise from thehigh thermal conductivity of both Cu and graphene.

V. CONCLUSIONS

The thermal conductivity of electrochemically depositedCu-gr composite and copper films on OFHC copperfoils has been determined by the 3x method andmultilayer analysis. The value of thermal conductivityof electrochemical deposited Cu has been found to be380 W/m.K at 300 K (27 �C) and thus lower than thatof OFHC Cu because of the oxygen introduced from theimpurities in the CuSO4 electrochemical depositionbath. In contrast, the thermal conductivity of Cu-grcomposite films is increased to 460 W/m.K at 300 K(27 �C) by incorporation of graphene through suspen-sion in the electrolytic bath. The thermal conductivity ofCu-gr films measured in the temperature range of 250 to350 K (–23 �C to 77 �C) showed the expected variation,with a higher value of 510 W/m.K at 250 K (–23 �C)and a lower value of 440 W/m.K at 350 K (77 �C). Thevolume fractions of graphene determined from quanti-tative metallography and electrical resistivity measure-ments were used to determine the volume fraction that iscloser to the actual value.EMA is used to determine the interface thermal

conductance between Cu and graphene from the averagesize of the graphene platelets and thermal conductivityof Cu and graphene. Electron–electron couplingbetween Cu and graphene using diffuse mismatch modeland electron–phonon coupling within graphene is foundto give higher value of the interface thermal conduc-tance and the value was found to be closer to that

Table VI. Different Components of Thermal Conductance Across Cu and Graphene Interface*

Electron–Phonon andPhonon–Phonon Coupling

Total ThermalConductance

Electron–Electron andElecton–Phonon Coupling

Total ThermalConductance

hep (Cu) hpp (Cu-graphene) h hee (Cu- graphene) hep (graphene) h

6.9 22.6 (AMM) 5.3 1860 (DMM) 13.8 13.76.9 15.0 (DMM) 4.76.9 18.9 (DMM) 5.0

*The first model is based on phonon–phonon coupling and the second model is based on electron–electron coupling between Cu and graphene.All values of thermal conductance are given in units of 108 W/m2.K. The values obtained using the AMMs and the DMMs are listed for hpp.


determined from EMA. It is concluded that the interfacethermal conductance, close to 1.4 GW/m2.K, is not thelimiting factor to achieve a greater improvement in thethermal conductivity of the Cu-gr composites. Instead,the smaller size of graphene platelets with a resultantlower average value of thermal conductivity of grapheneis thought to be the limiting factor. In addition, thickergraphene platelets with more than three atomic layersare expected to possess higher thermal conductivity inthe presence of matrix so that the quenching of phononsthat carry the heat in graphene is confined only to theouter layers.

ACKNOWLEDGMENT

This research is supported by National ScienceFoundation Grant CMMI #1049751.

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Copper-graphene Composite Heat Sink