Copper-graphene Composite Heat Sink
Content
Thermal Conductivity of Copper-Graphene Composite Films Synthesized by Electrochemical Deposition with Exfoliated Graphene Platelets
K. JAGANNADHAM Samples of graphene composites with matrix of copper were prepared by electrochemical codeposition from CuSO4 solution with graphene oxide suspension. The thermal conductivity of the composite samples with different thickness and that of electrodeposited copper was determined by the three-omega method. Copper-graphene composite films with thickness greater than 200 lm showed an improvement in thermal conductivity over that of electrolytic copper from 380 W/m.K to 460 W/m.K at 300 K (27 °C). The thermal conductivity of coppergraphene 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 the composite samples and determine the interfacial thermal conductance between copper and graphene. The values of interface thermal conductance greater than 1.2 GW/m2.K obtained from the acoustic and the diffuse mismatch models and from the EMA modeling of the experimental results indicate that the interface thermal resistance is not a limiting factor to improve the thermal conductivity of the copper-graphene composites. DOI: 10.1007/s11663-011-9597-z Ó The Minerals, Metals & Materials Society and ASM International 2011
I.
INTRODUCTION
LOW-COST manufacturing of thermal interface materials and heat spreaders with high capacity to dissipate thermal energy is of significant importance in nanoelectronics and high-frequency and high-power devices 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 higher compared with other materials. Of these, the measurements at room temperature showed that the thermal conductivity of free-standing graphene is the highest with the value between 3000 and 5000 W/m.K and, thus, greater than that of other forms of carbon discovered until now. However, it was also shown recently[9] that the thermal conductivity of monolayer of graphene supported on SiO2 is only 600 W/m.K as a result of phonon leaking and phonon scattering by the substrate. To alleviate the problems associated with low thermal conductivity thermal interface materials[10] (TIM), recently we studied a simple method of processing In-graphene and In-Ga-graphene composites[11,12] and found that the thermal conductivity at 300 K (27 °C) is improved by a factor of 2.5 and 3, respectively, compared with that of In and In-Ga. In general, a heat spreader system contains a high capacity heat spreader and an external cooling system in addition to the TIM attached to the active device region. In the current work,
K. JAGANNADHAM, Associate Professor, is with the Department of 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.
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we studied the preparation of copper-graphene (Cu-gr) composite heat spreaders by electrochemical codeposition from CuSO4 solution with graphene oxide suspension. The thermal conductivity of the Cu-gr composites was determined experimentally and modeled to determine the importance of interface thermal conductance and graphene particle size.
II.
EXPERIMENTAL PROCEDURES
The preparation of copper-graphene composite samples has been described at length in the previous work.[11,13] A brief description is provided here. Graphene oxide (GO) films were prepared by chemical exfoliation from microcrystalline graphite[14] supplied by Asbury Graphite Inc. (Asbury, NJ) The details of preparation of exfoliated GO are discussed at length in our previous work.[11] A suspension of GO particulates in a 0.2 M solution of CuSO4 was prepared. Electrochemical codeposition from a bath containing graphene oxide (GO) suspension in a solution of technical grade CuSO4 in distilled water was carried out on oxygen-free high-conductivity (OFHC) copper foils. The pH was maintained close to 7, and a low current density of 1.75 mA/cm2 and growth rate of 2 to 3 lm/h[13] were used with a pure Cu anode to achieve smooth films. GO is hydrophobic on the basal plane and hydrophilic at the edges so that the suspension of the GO particulates is maintained 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 to smaller size of approximately 8-mm 9 10-mm area using a diamond saw. To evaluate the improvement in
METALLURGICAL AND MATERIALS TRANSACTIONS B
the thermal conductivity introduced by graphene, samples of electrolytic Cu were also deposited on the Cu foil from the same electrolytic bath but without GO suspension. The thickness of OFHC Cu foil in all the samples was 135 lm, but the thickness of the deposited composite film on the opposite sides of the Cu foil varied. The electrolytic Cu and Cu-gr samples were heated in flowing hydrogen atmosphere at partial pressure of 20 Torr and 673 K (400 °C) for 3 hours to reduce GO to graphene and Cu2O or CuO to Cu and thus form Cu-gr composites. Additional characterization by scanning electron microscopy (SEM) for graphene morphology and distribution of graphene platelets, energy dispersive spectrometry (EDS) for presence of oxygen and other impurities, and electrical conductivity as a function of temperature were carried out.[13] The electrical resistivity and temperature coefficient of resistance (TCR) were used previously to determine the volume fraction and resistivity of thinner films of graphene in all the samples.[13] SEM imaging combined with lineal fraction analysis in quantitative metallography[16] was used to determine the volume fraction of thicker graphene in the Cu-gr samples. Thermal conductivity of the samples was measured using the 3-x method. The details of the 3-x method are described at length in the previous work[12] and in the original Reference 17. A schematic diagram illustrating the sample with a gold heater line on the surface is presented in Figure 1. In the 3-x method, a line heater is used both as a source of heat and to measure the temperature on the surface of the sample for which the thermal conductivity is measured.[12,17] The Cu-gr composite sample surface is isolated by spreading an electrically insulating polymer film. The polymer film is a commercial thermosetting white epoxy S-30/3045 supplied by Devcon (Ellsworth Adhesives, Germantown, WI). In addition, thin films of insulating Si and ZrO2 each with thickness approximately 0.2 lm were deposited by laser physical vapor deposition (LPVD) on the top of the polymer film to enable the adhesion of a gold heater line to the surface. The gold heater line, with a width below 200 lm and length up to 8 mm, was deposited by LPVD using a negative stainless steel mask placed on the surface. Indium foils, with a width of 1 mm and length of 5 mm, were pressed on to either end of the gold heater line to make electrical contacts using
gold wires of 0.1 mm diameter. Two contacts on the opposite sides of the heater line were used to input the current and the other two were used to measure the output of voltage. The samples were mounted using ZnO paste on the substrate stage of a temperature-controlled vacuum chamber supplied by MMR Technology (Mountain View, CA). The chamber was kept under vacuum at 6 millitorr using a mechanical pump during electrical measurements. The temperature of the sample stage was controlled to ± 0.1 K by K-20 controller in conjunction with high-pressure nitrogen gas that was passed through a nozzle in the stage system of the chamber or by the Joule-Thomson effect. The measurements were made at three 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 was reached, the internal source of the lock in amplifier was used to supply power at the set input voltage V1 and frequency f, and the output voltage V3 was measured at frequency 3f. SR 830 DSP lock in amplifier supplied by Stanford Research Systems (Sunnyvale, CA) was used to supply the power at frequency f and measure the voltage at frequency 3f. The measurements were made as a function of frequency f (2pf = x) that was increased from 1 Hz to 3.5 kHz in different steps to cover the complete range. The electrical resistance (R) and temperature coefficient of resistance (a = dR/RdT) of the heater line were measured from the four contacts, with the two outer contacts to supply current at a constant value and the other two to measure the voltage. Keithley source meter 2400-LV and the nanovoltmeter 2182 (Keithley, Cleveland, OH) were used with reverse current procedure to cancel out the thermal electromotive force at the contacts. The average of the absolute value of the voltage was used to determine the resistance using Ohm’s law. The temperature was varied from 290 K to 315 K (17 °C to 42 °C) in steps 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 =ðaV1 Þ ½1
Fig. 1—Schematic illustration of the sample for the 3-x measurements. The sample with electrically insulating films on the top and the gold heater with indium foils attached on either side are shown. The gold wire electrical contacts on either side of the heater are also shown.
METALLURGICAL AND MATERIALS TRANSACTIONS B
where V1 is the voltage at frequency f, V3 at frequency 3f, and a is the temperature coefficient of resistance of the heater line. The power input per unit length into the heater was determined from V12/(lR) where l is the length of the heater line. A graph of dT per unit power input was generated as a function of ln(f) over the complete range of frequency at a given substrate temperature. These results were curve fitted using a multilayer analysis[18] with the help of a computer program. The parameters in the multilayer analysis consist 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. Interfacial thermal resistance between layers was also included in the multilayer analysis by introducing a layer with small thickness, heat capacity, and finite thermal conductivity. The multilayer analysis was started using the known values of thermal conductivity and heat capacity of
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different layers and the thickness of different layers shown schematically in Figure 2. The values of the parameters were modified during the numerical analysis to fit the experimental data and thus determine the thermal conductivity of Cu-gr layers. The polymer film had low thermal conductivity. The thermal waves penetrated into the ZnO paste only when the polymer film was thin. Therefore, an additional set of measurements were also made on two samples with thinner insulating polymer film. The thicknesses of the polymer film and the insulating layer are given in the results of curve fitting described subsequently.
III.
EXPERIMENTAL RESULTS
A. Results of Characterization of the Cu-gr Composite Films An SEM image of the cross section of the composite sample is provided in Figure 3 along with the energy dispersive spectrometry of the graphene film in Cu in Figure 4. The SEM image in backscattering mode and the X-ray maps of the carbon and the Cu signals are shown in Figure 3. The labeling in Figure 3 shows the pure Cu foil in the center and the Cu-gr composite films outside. The image also contains some graphene particulates that were sticking on the surface of the pure Cu film in the center as a result of contamination from cutting with diamond saw. It is observed from this image that graphene is distributed uniformly in the composite films outside the Cu foil. The X-ray map of carbon signal is shown in the middle of Figure 3.
The carbon signal in the center region of Cu is to the result of instrumental noise. If the noise from the signal is taken into account, then the middle region of Cu has no carbon but the outer regions of Cu-gr have a higher carbon signal. The Cu signal shown on the right-hand side of Figure 3 contains strong Cu signal from the center and a reduced Cu signal from Cu-gr composite. The EDS spectrum obtained from graphene particulates in the Cu-gr film is shown in Figure 4. The spectrum contains strong C and Cu peaks and a small O peak. The oxygen peak is associated with residual concentration in the film. X-ray diffraction[13] from a large area of the Cu-gr films in the as-deposited condition showed a diffraction peak at 2h = 11.8 deg that is associated with graphene oxide particulates but did not show any peak associated with graphite at 2h = 26.5 deg. This result indicated that the composite samples did not contain any graphite. However, the GO peak was not present on annealing in hydrogen at 673 K (400 °C) for 3 hours, which indicated that GO is converted to graphene layers of small thickness. Transmision electron microscopy[13] of the suspended GO particulates on microscopy grid also showed thin films with a diffraction pattern containing diffraction spots arranged in a hexagonal pattern. These results presented in the previous work[13] and the SEM characterization
Fig. 2—Schematic illustration of the different layers in the cross section of the composite samples prepared for thermal conductivity measurement. The gold heater wire and indium contacts are not shown.
Fig. 4—EDS spectrum of graphene particulates on Cu matrix in Cu-gr composite films. The residual oxygen peak is observed in addition to the major carbon and Cu peaks.
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 the left. 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 on the 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 the Cu foil region in the center were present because of contamination during cutting with the diamond wheel.
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confirmed that the composite films contained graphene platelets in a Cu matrix. B. Results of Thermal Conductivity A list of all the samples and the volume fractions of graphene evaluated from the measurement of electrical resistivity[13] and quantitative metallography are provided in Table I. First, the results obtained on the thermal conductivity of two samples of Cu-gr4 using a thinner and a thicker polymer insulating film on the top are described. Next, the results obtained on all the samples are presented along with the temperature dependence. The results obtained at 300 K (27 °C) from sample Cu-gr4-1, which consisted of Cu-gr composite films of thickness 240 lm deposited on the top and 205 lm at the bottom of the Cu foil are shown in Figure 5. The thermal conductivity, heat capacity, and thickness of the different layers used in the multilayer analysis are listed in Table II. The heat capacity of Cu and ZnO are taken from Reference 19. The thickness of the polymer film was 27.5 lm and that of the insulating ZrO2, Si, and the polymer was 11.8 lm. Thus, the total thickness of the top insulating films was 39.3 lm. An interfacial layer of 0.01 lm was used to represent the thermal resistance between the gold heater line and the top of ZrO2 film. The thermal waves penetrated to the bottom ZnO layer used for mounting the sample on the substrate holder in the chamber. The region of the graph with smaller slope in Figure 5 with 0 < ln(f) < 5 represents the bottom layers consisting of Cu-gr, Cu, Cu-gr, and ZnO. The region with the higher slope with 5 <ln(f) < 8.2 represents the top layers of electrically insulating films. The results obtained at 300 K (27 °C) from sample Cu-gr4-2, which consisted of Cu-gr composite film of thickness 215 lm deposited on the top and 165 lm at the bottom of the Cu foil, are shown in Figure 6. The thermal conductivity, heat capacity and thickness of the different layers used in the multilayer analysis are listed in Table III. The thickness of the polymer film was 66.1 lm and that of the insulating ZrO2, Si, and polymer layer was 13.2 lm. Thus, the total thickness
of the top insulating films was 79.3 lm, which is larger than that in the sample Cu-gr4-1. As stated, an interfacial layer of thickness 0.01 lm was used to represent the thermal resistance between the gold heater line and the ZrO2 film. The thermal waves have not
Fig. 5—Blue diamond symbols represent the experimental values of dT per unit power input and red squares represent the results obtained from multilayer analysis for sample Cu-gr4-1 with thinner polymer 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 the Experimental 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* Thermal Heat Conductivity Capacity Thickness (102 W/m.K) (106 J/m3.K) (lm) 0.0010 0.0060 0.0503 4.6 3.9 4.6 1.0 0.001 2.0 2.0 3.5 3.5 3.5 2.96 0.01 11.8 27.5 240 135 205 1200
Identified Layer Interface layer ZrO2, Si and polymer Polymer layer Cu-gr Composite Cu foil Cu-gr Composite ZnO
*The results of curve fitting are shown in Fig. 5. The polymer film plus the insulating layer on the top is relatively thin (39.3 lm). The temperature of the substrate was 300 K (27 °C).
Table I.
List of Samples Tested with Values of the Volume Fraction of Graphene fg Evaluated from Electrical Resistivity and Metallography* fg Thermal Conductivity (102 W/m.K) Metallography 0 0 0.25 0.24 0.26 0.19 0.20 Total Thickness (lm) 135 425 250 445 395 385 432 T = 250 K (–23 °C) 4.2 4.0 — 5.1 — — T = 300 K (27 °C) 3.9 3.8 4.5 4.6 4.6 4.6 4.6 T = 350 K (77 °C) 3.8 3.7 — 4.4 — — —
Sample Cu foil Cu-elec Cu-gr3 Cu-gr4 Cu-gr5 Cu-gr6 Cu-gr7
Resistivity 0 0 0.08 0.09 0.11 0.11 0.09
*The values of thermal conductivity of OFHC Cu foil, electrolytic Cu (Cu-elec), and Cu-gr composite samples determined by the 3-x method and analyzed by multilayer analysis are shown. The values of thermal conductivity are accurate to ±0.1. Cu-elec is electrolytic Cu Deposited on OFHC Cu foil of thickness 135 lm.
METALLURGICAL AND MATERIALS TRANSACTIONS B
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Fig. 6—The blue diamond symbols represent the experimental values of dT per unit power input. The red circles represent the results obtained from multilayer analysis for sample Cu-gr4-2 with thicker polymer film. The temperature of the substrate was 300 K (27 °C). Table III shows the different parameters used in the multilayer analysis.
Table III. Values of the Different Parameters used to Fit the Experimental 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* Thermal Heat Conductivity Capacity Thickness (102 W/m.K) (106 J/m3.K) (lm) 0.0010 0.00295 0.0070 4.6 3.9 4.6 0.001 2.0 2.0 3.5 3.5 3.5 0.01 13.2 66.1 215 135 165
film provided on the surface. Different parameters used in the analysis and the curve-fitted graphs are not presented in this article for the sake of brevity. The results were found to be the same from both samples, and the 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 the results are shown in Table I. The thermal conductivity of Cu-gr film in sample Cu-gr3 was found to be slightly lower than that of other samples, and this result is attributed to slightly variable microporosity introduced during electrochemical deposition. The temperature dependence of the thermal conductivity of the samples Cu-gr4-1 and Cu-elec with thinner polymer-insulating film on the surface was determined by 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 multilayer analysis. The values of the different parameters used in the multilayer analysis and the curve fitted graphs are not presented for the sake of brevity. The values of the thermal conductivity obtained for the Cu-gr films are shown in Table I.
Identified Layer Interface layer ZrO2, Si and Polymer Polymer layer Cu-Gr Composite Cu foil Cu-Gr Composite
IV.
DISCUSSION
*The results of curve fitting are shown in Fig. 6. The polymer film plus the insulating layer on the top is relatively thick (79.3 lm). The temperature of the substrate was 300 K (27 °C).
penetrated into the ZnO layer used for mounting the sample on the substrate holder because the polymer layer with lower thermal conductivity has lower thermal diffusion depth. The region of the graph with smaller slope in Figure 6 with 0 < ln(f) < 1 represents the bottom layers consisting of Cu-gr, Cu, and Cu-gr. The region with higher slope with 1 < ln(f) < 8.2 represents the top insulating layers. The multilayer analysis of the Cu-gr4-1 and Cu-gr4-2 samples with a different thickness of polymer film was carried out so that the larger region of smaller slope, as shown in Figure 5, could be curve fitted with more confidence. However, the values of the thermal conductivity of the Cu-gr layers shown in Tables II and III are same indicating that the analysis in Figure 6 is equally valid. The thickness of the Cu-gr composite film in both these samples is large (>200 lm), so that the thickness dependence of thermal conductivity did not change the value of thermal conductivity obtained in these two samples significantly. A multilayer analysis was also performed on the 3-x experimental data obtained from two samples of electrolytic copper (Cu-elec) film deposited on the Cu foil but with thinner and thicker electrically insulating polymer
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The results presented in Table I show that the thermal conductivity 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 Cu foil on which it is deposited. The slightly lower value is associated with a higher electrical resistivity[13] that is also observed in the electrodeposited Cu. An EDS analysis[13] of the samples showed presence of higher oxygen level in all the electrodeposited samples, and it is attributed to the impurities in the technical grade CuSO4 used in the electrochemical deposition. The details of the impurities and the higher electrical resistivity observed in the electrodeposited films are presented in the previous work.[13] Therefore, a subsequent comparison of thermal conductivity of Cu-gr films will be made with respect to the electrodeposited Cu film instead of the OFHC 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 smaller value of 4.5 W/cm.K in the sample Cu-gr3 with smaller total thickness of the composite film, as shown in Table I. This result indicates a large improvement from 3.8 W/cm.K associated with electrolytic Cu. The temperature dependence of thermal conductivity of the Cu-gr film in the sample Cu-gr4-1 is similar to that of electrolytic Cu in the temperature range of 250 K to 350 K (–23 °C to 77 °C). It is lower at higher temperature[20,21] as a result of increased scattering of electrons in Cu and phonons in graphene. These results are analyzed using effective mean field approximation (EMA). In particular, EMA is used to determine the dependence of thermal conductivity on the volume fraction of graphene incorporated in the Cu-gr composite films and to evaluate the interface thermal conductance between Cu and graphene.
METALLURGICAL AND MATERIALS TRANSACTIONS B
A. Evaluation of Volume Fraction of Graphene The volume fraction of graphene determined from a lineal fraction analysis using quantitative metallography[16] is presented in Table I. SEM imaging of graphene in backscattering mode showed only those graphene platelets that are thicker exhibit dark contrast. As a result, thinner graphene platelets such as single or bilayer graphene that exhibit weak contrast cannot be identified. Therefore, the volume fraction estimated from quantitative metallography is slightly lower than the actual value because the graphene films of smaller thickness are not included. The volume fraction of graphene determined from electrical resistivity and temperature coefficient of electrical resistance (TCR)[13] is also presented in Table I. The two unknowns, resistivity and the volume fraction of graphene, were solved using resistivity and the TCR of the composite samples in terms of the formulations of EMA.[13] Although this method is helpful to evaluate the volume fraction of thinner graphene platelets that contribute significantly to electrical conductivity, thicker graphene platelets that do not contribute to improvement cannot be evaluated. The electrical conductivity of graphene platelets thicker than 5 to 10 atomic layers approaches the characteristics of graphite that is semimetallic with lower electrical conductivity. Therefore, thicker graphene platelets do not contribute to electrical conductivity. However, both thinner and thicker graphene platelets contribute significantly to thermal conductivity[22–24] because thermal conductivity in the basal plane of graphene platelet is much higher than that of Cu. Therefore, the volume fraction of graphene that should be included in the analysis of thermal conductivity of Cu-gr composites is higher than the value obtained from quantitative metallography. The volume fraction of thinner and thicker graphene platelets in the sample is determined by adding the values obtained from electrical resistivity and quantitative metallography, respectively. The volume fraction of graphene determined by electrical conductivity is much smaller and within the accuracy achieved by quantitative metallography, and therefore, the total volume fraction is 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 thermal conductivity of indium-graphene composites and carbon nanotube composites.[12,25] Therefore, the formulations will be described only briefly. Under this approximation, the isotropic thermal conductivity of the Cu-gr composite medium is given by È À Á É Kc ¼ Kg Km 3 À 2fg þ 2fg Kg È À ÁÉ ½2 = fg Km þ Kg 3 À fg þ afg where fg is the volume fraction of graphene and K stands for thermal conductivity, with subscript c for the composite (Cu-gr), g for graphene, and m for matrix or Cu. The parameter a is given by Km/hr where h is the interfacial thermal conductance between the matrix and
METALLURGICAL AND MATERIALS TRANSACTIONS B
graphene and r is the effective radius of the graphene platelets. In the current analysis, r is taken as half the average size of graphene platelets present in the composite samples. EMA is used to determine the interfacial thermal conductance, h from Eq. [2]. The experimental determinations[8,22] of Kg have shown that its value is high when the size of the graphene platelets is bigger, such as 5 lm. A modeling analysis[21,23] also indicates that the value is expected to be in the experimentally observed range. Modeling[24] showed also that the value of Kg decreases to 1000 W/m.K when the size of the graphene platelet is 1 lm. The size (L) dependence of thermal conductivity in graphene nanoribbons has been shown[25] to follow (L)b, where b remains between 0.35 and 0.45. In addition, Klemens pointed out[26,27] that thermal energy leaks in to the matrix when graphene is present on a substrate, thereby, the thermal conductivity will be reduced further by 20 to 50 pct if the phonon velocity in the matrix is smaller than that in graphene. This result has been confirmed recently from the thermal conductivity measurements of graphene suspended on SiO2.[9] The average size Lav of the graphene platelets and the weighted average of the thermal conductivity Kg were determined from the distribution of the platelet sizes in the graphene suspension.[12] The same graphene suspension is used in the current experiments so that the average size of graphene platelets is 0.64 lm and the average value of Kg is 800 W/m.K. EMA is performed with these values of the parameters and in particular a = Km/hr where r = 0.32 lm. The results of interfacial thermal conductance h obtained from Eq. [2] are shown in Table IV. The results shown in Table IV illustrate that the interface thermal conductance remains between 1.2 and 2.2 in units of 109 W/m2.K, which is reasonably high. The variation in h between the samples is caused by the difficulty to determine the volume fraction of graphene accurately in the composites. C. Interface Thermal Conductance from Acoustic and Diffuse Mismatch Models The thermal conductivity in Cu is electron mediated, whereas that in graphene is phonon mediated in the temperature range of 250 K to 350 K (–23 °C to 77 °C).
Table IV. Value of Interfacial Thermal Conductance h Determined 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* Composite Sample Cu-gr3 Cu-gr4 Cu-gr5 Cu-gr6 Cu-gr7 Kc (102 W/m.K) 4.5 4.6 4.6 4.6 4.6 fg 0.33 0.33 0.37 0.30 0.29 h (109 W/m2.K) 1.2 1.5 1.3 1.9 2.2
*fg in this table is the sum of the two values of the volume fraction shown in Table I. See the explanation in Section IV.
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Table V. Parameters used in the Calculation of the Interface Thermal Conductance* Velocity (103 m/s) Medium Cu Graphene Density (106 gm/m3) 8.93 2.26 Longitudinal vl 4.8 24.0 Transverse vt 2.4 16.0 Debye vd 2.8 18.6 Lattice Specific Heat (106 J/m3.K) 3.5 1.84 c (103 J/m3.K2) 0.097 0.008 vf (106 m/s) 1.57 1.10
*Cu is treated as three-dimensional medium (3/v2 = 1/v2 + 2/v2) and graphene as two-dimensional medium (2/v2 = 1/v2 + 1/v2) for calculating d l t d l t the 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. The values of c and the Fermi velocity vf are taken from Refs. 29 and 30 for Cu and from Refs. 31 and 32 for graphene.
The interface thermal conductance is evaluated previously[12] using two approaches. In the first, we consider electron–phonon coupling in Cu and phonon–phonon coupling between Cu and graphene. Next, electron– electron coupling between Cu and graphene and electron–phonon coupling in graphene are used in the second approach. The electron–phonon coupling term hep in Cu is determined using[28] À Á1=2 Gep ¼ Cem =s and hep ¼ Gep Kpm ½3 where Cem is the electronic specific heat of Cu and s is the relaxation time for electron–phonon energy transfer. The values of different parameters used in the modeling are provided in Table V.[29–32] The electronic specific heat is calculated using the Cem = cT where c = 97.0 J/m3.K from Table V.[29] The relaxation time is assumed to be 1 ps[28] so that Gep = 29.1 9 106 GW/m3 at 300 K (27 °C). The phonon thermal conductivity of Cu is calculated from Kpm = Clmvdmlp/3 where Clm is the lattice 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 conductance from electron–phonon coupling hep is 6.87 9 108 W/m2.K. The phonon–phonon contribution hpp to the interface thermal conductance is determined from the acoustic or the diffuse mismatch models.[33,34] In the acoustic mismatch model[33] hpp ¼ Clm vdm a=4 ½4 where a is the transmission coefficient determined from À Á2 ½5 a ¼ 4Zm Zg = Zm þ Zg where Z is the acoustic impedance of each medium given by the product of density and Debye velocity, and the subscripts m and g refer to Cu and graphene, respectively. The value of hpp from acoustic mismatch model is determined using the values of parameters listed in Table V and found to be 2.26 9 109 W/m2.K. The expression for hpp in the diffuse mismatch model[33] is n À o Á hpp ¼ 1:02 Â 1010 T3 R 1=v2 R ð1=v2 m g n o 2 2 = R ð1=vm þ 1=vg ½6 In Eq. [6], the summation is performed over the longitudinal and the two transverse modes in Cu but
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only over the longitudinal and the single transverse mode in graphene because it is a two-dimensional medium. Using the parameters given in Table V, the value of hpp is found to be 1.50 9 109 W/m2.K. The net thermal conductance[28] determined from 1/h = 1/hep + 1/hpp is 4.7 9 108 W/m2.K from the diffuse mismatch model (DMM) and 5.3 9 108 W/m2.K from the acoustic mismatch model (AMM). An alternative expression from diffuse mismatch model[34] is, À Á ½7 hpp ¼ Clm vdm Clg vdg =4 Clm vdm þ Clg vdg where C represents, as before, the lattice specific heat and vd the Debye velocity. Using the parameters provided in Table V, the value of hpp = 18.9 9 108 W/m2.K. In this case, the total thermal conductance is found to be 5.0 9 108 W/m2.K. From the preceding results, the values of h from the acoustic and the diffuse mismatch models are close. In the next representation, the total thermal conductance will consist of electron–electron coupling between Cu and graphene and electro–phonon coupling in graphene. The first part is determined from,[35] À Á hee ¼ Ceg vfg Cem vfm T=4 Ceg vfg þ Cem vfm ½8 where vfg represents the Fermi velocity in graphene and vfm in Cu, Ceg is the electronic specific heat of grapheme, and Cem is that of Cu. The electronic specific heat is determined as before from Ce = cT and the values of c are listed in Table V for each medium so that the value of hee at T = 300 K (27 °C) is found to be 18.6 9 1010 W/m2.K. The electron–phonon coupling term[28] in graphene hep is determined as before from hep = {Gep Kpg}1/2 where Gep = (Ceg/s) and Kpg is the phonon thermal conductivity in graphene. Using the parameters listed in Table V, Gep = 2.38 9 106 GW/m3.K. The phonon thermal conductivity Kpg in graphene is size dependent. The distribution of graphene platelet sizes in the suspension is evaluated, and using the size dependence[25] of thermal conductivity, the average value was shown previously[12] to be 800 W/m.K so that hep = 1.38 9 109 W/m2.K. The net thermal conductance h given by 1/h = 1/hee + 1/hep is found to be 1.37 9 109 W/m2.K. Therefore, the value of thermal conductance based on electron–electron coupling is higher than when phonon–phonon coupling is used between Cu and graphene. The different values obtained for the interface thermal conductance from the various components and the total
METALLURGICAL AND MATERIALS TRANSACTIONS B
thermal conductance using the two approaches of carrier coupling are listed in Table VI. Although the electron– phonon coupling term in graphene is weak, it is stronger than in Cu by a factor of two. Similarly, the DMM indicates that electron–electron coupling term is stronger than phonon–phonon coupling term between Cu and graphene. The value of the interface thermal conductance, from Table VI, is lower than the experimental value shown in Table IV when phonon–phonon coupling is considered between Cu and graphene. The value obtained from electron–electron coupling, as shown in Table VI, is closer to the experimentally determined value using EMA. The lower Debye velocity of sound in Cu compared with that in graphene by almost a factor of 7, as shown in Table V, indicates clearly that quenching of phonons in graphene will take place.[27] Therefore, the thermal conductivity Kg of single- and double-layer graphene platelets will be reduced both from quenching of phonons and smaller size of graphene platelets. From this point of view, graphene platelets of 4 to 10 atomic layers thick wherein the phonon transport within inner layers is still effective seem to be advantageous. It is important to determine the thickness dependence of thermal conductivity of graphene platelets, especially when in contacts with a matrix such as Cu. It is useful to compare the current results with that of interface thermal conductance between metals and graphite obtained experimentally[36,37] and using modeling.[38] Although, these studies did not examine directly the interface thermal conductance between Cu and graphite, the results obtained for Al, Au, Ti, and Cr on C-axis-oriented graphite are relevant. The results showed that the interface thermal conductance remained between 30 and 100 MW/m2.K with lower range values applicable to noninteracting face-centered cubic metals like Au and higher values for interacting metals like Ti and Cr. The larger value of 1.0 9 108 W/m2.K in these measurements and modeling using phonon coupling[36,37] is only smaller by a factor of 5 from the DMM approach used in the current work. The main discrepancy is associated with the C-axis orientation of graphite crystals on which these studies were performed. The thermal conductivity of a few layers of graphene is three orders of magnitude smaller in the C-direction compared with the value in the basal plane or normal to the C-direction. The values shown for the present work assume thermal transport in the basal plane of graphene platelets, and hence, the interface thermal conductance is also expected to be much higher.
Table VI.
The modified DMM approach with phonon coupling used in Reference 38 estimates the interface thermal conductance between basal plane graphite and Al at 300 K (27 °C) to be close to 4 9 108 W/m2.K. This value is close to that obtained in the current calculations using phonon–phonon coupling in the DMM approach. The thermal conductivity of Cu is a factor of 2 higher than that of Al and that of graphene is also higher than that of graphite so that the interface thermal conductance between Cu and graphene is expected to be larger than that between Al and graphite. More importantly, the thermal conductance using electron–electron coupling seems to be higher than that using phonon– phonon coupling between Cu and graphene. The higher experimental value of interface thermal conductance in the current analysis using EMA could arise from the high thermal conductivity of both Cu and graphene.
V.
CONCLUSIONS
The thermal conductivity of electrochemically deposited Cu-gr composite and copper films on OFHC copper foils has been determined by the 3x method and multilayer analysis. The value of thermal conductivity of electrochemical deposited Cu has been found to be 380 W/m.K at 300 K (27 °C) and thus lower than that of OFHC Cu because of the oxygen introduced from the impurities in the CuSO4 electrochemical deposition bath. In contrast, the thermal conductivity of Cu-gr composite films is increased to 460 W/m.K at 300 K (27 °C) by incorporation of graphene through suspension in the electrolytic bath. The thermal conductivity of Cu-gr films measured in the temperature range of 250 to 350 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). The volume fractions of graphene determined from quantitative metallography and electrical resistivity measurements were used to determine the volume fraction that is closer to the actual value. EMA is used to determine the interface thermal conductance between Cu and graphene from the average size of the graphene platelets and thermal conductivity of Cu and graphene. Electron–electron coupling between Cu and graphene using diffuse mismatch model and electron–phonon coupling within graphene is found to give higher value of the interface thermal conductance and the value was found to be closer to that
Different Components of Thermal Conductance Across Cu and Graphene Interface* Total Thermal Conductance h 5.3 4.7 5.0 Electron–Electron and Electon–Phonon Coupling hee (Cu- graphene) 1860 (DMM) hep (graphene) 13.8 Total Thermal Conductance h 13.7
Electron–Phonon and Phonon–Phonon Coupling hep (Cu) 6.9 6.9 6.9 hpp (Cu-graphene) 22.6 (AMM) 15.0 (DMM) 18.9 (DMM)
*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.
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determined from EMA. It is concluded that the interface thermal conductance, close to 1.4 GW/m2.K, is not the limiting factor to achieve a greater improvement in the thermal conductivity of the Cu-gr composites. Instead, the smaller size of graphene platelets with a resultant lower average value of thermal conductivity of graphene is thought to be the limiting factor. In addition, thicker graphene platelets with more than three atomic layers are expected to possess higher thermal conductivity in the presence of matrix so that the quenching of phonons that carry the heat in graphene is confined only to the outer layers. ACKNOWLEDGMENT This research is supported by National Science Foundation Grant CMMI #1049751.
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1. E.M. Garmire and M.T. Tavis: IEEE J. Quant. Electron., 1984, vol. QE-20, pp. 1277–80. 2. J. Piprek, J.K. White, and A.J. Spring Thorpe: IEEE J. Quant. Electron., 2002, vol. 38, pp. 1253–59. 3. V.O. Turin: Electron. Lett., 2004, vol. 40, pp. 81–83. 4. G.A. Slack: J. Appl. Phys., 1964, vol. 35, pp. 3460–66. 5. G.A. Slack: Phys. Rev., 1962, vol. 127, pp. 694–701. 6. P. Kim, L. Shi, A. Majumdar, and P.L. McEuen: Phys. Rev. Lett., 2001, vol. 87, pp. 215502-1–4. 7. A.A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweidebrhan, F. Miao, and C.N. Lau: Nano Lett., 2008, vol. 92, pp. 151911-1–3. 8. A.A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C.N. Lau: Nano Lett., 2008, vol. 8, pp. 902–07. 9. J.H. Seol, I. Jo, A.L. Moore, L. Lindsay, Z.H. Aitken, M.T. Pettes, X. Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R.S. Ruoff, and L. Shi: Science, 2010, vol. 328, pp. 213–16. 10. R. Prasher: Proc. IEEE, 2006, vol. 94, pp. 1571–87. 11. A.N. Sruti and K. Jagannadham: J. Elect. Mater., 2010, vol. 39, pp. 1268–76. 12. K. Jaganandham: J. Elect. Mater., 2011, vol. 40, pp. 25–34. 13. K. Jagannadham: J. Electrochem. Soc., in press. 14. S. Stankovich, D.A. Dikin, R.D. Piner, K.M. Kohlhaas, A. Kleinhammes, Y. Jia, Y. Wu, S.T. Nguyen, and R.S. Ruoff: Carbon, 2007, vol. 45, pp. 1558–65.
15. J. Kim, L.J. Cote, F. Kim, W. Yuan, K.R. Shull, and J. Huang: J. Amer. Chem. Soc., 2010, vol. 132, pp. 8180–86. 16. E.E. Underwood: Applications of Quantitative Metallography, Mechanical Testing, Metals Handbook, 8th ed., vol. 8, ASM, Materials Park, OH, 1973, p. 37. 17. D.G. Cahill: Rev. Sci. Instrum., 1990, vol. 61, pp. 802–08. 18. J.H. Kim, A. Feldman, and D. Novotny: J. Appl. Phys., 1999, vol. 86, pp. 3959–63. 19. Y.S. Touloukian and E.H. Buyco: Thermophysical Properties of Matter, The TPRC Data Series, Specific Heat of Metallic Elements and Alloys, vol. 4, and Specific Heat of Nonmetallic Solids, vol. 5, IFI/Plenum, New York, NY, 1970. 20. J. Yang: in Thermal Conductivity: Theory, Properties, and Applications, T.M. Tritt, ed., Kluwer Academic/Plenum Press, New York, NY, 2004, p. 1. 21. S. Ghosh, D.L. Nika, E.P. Pokatilov, and A.A. Balandin: New J. Phys., 2009, vol. 11, pp. 095012-1–19. 22. S. Ghosh, I. Calizo, D. Teweldebrhan, E.P. Pokatilov, D.L. Nika, A.A. Balandin, W. Bao, F. Miao, and C.N. Lau: Appl. Phys. Lett., 2008, vol. 92, pp. 151911-1–3. 23. D.L. Nika, E.P. Pokatilov, A.S. Askerov, and A.A. Balandin: Phys. Rev. B, 2009, vol. 79, pp. 155413-1–12. 24. D.L. Nika, S. Ghosh, E.P. Pokatilov, and A.A. Balandin: Appl. Phys. Lett., 2009, vol. 94, pp. 203103-1–3. 25. Z. Ghuo, D. Zhang, and X.G. Gong: Appl. Phys. Lett., 2009, vol. 95, pp. 163103-1–3. 26. P.G. Klemens: Int. J. Thermophysics, 2001, vol. 22, pp. 265–75. 27. P.G. Klemens and D.F. Pedraza: Carbon, 1994, vol. 32, pp. 735– 41. 28. A. Majumdar and P. Reddy: Appl. Phys. Lett., 2004, vol. 84, pp. 4768–71. 29. D.L. Martin: Phys. Rev. B, 1973, vol. 8, pp. 5357–60. 30. A.C. Anderson and R.E. Peterson: Phys. Lett., 1972, vol. 38A, pp. 519–20. 31. R. Viana, H. Godfrin, E. Lerner, and R. Rapp: Phys. Rev. B, 1994, vol. 50, pp. 4875–77. 32. R.S. Deacon, K.C. Chuang, R.J. Nicholas, K.S. Novoselov, and A.K. Gein: Phys. Rev. B, 2007, vol. 76, pp. 08140 6-1–4. 33. E.T. Swartz and R.O. Pohl: Rev. Mod. Phys., 1989, vol. 33, pp. 605–68. 34. A. Minnich and G. Chen: Appl. Phys. Lett., 2007, vol. 91, pp. 073105-1–3. 35. B.C. Gundrum, D.G. Cahill, and R.S. Averback: Phys. Rev. B, 2005, vol. 72, pp. 245426-1–5. 36. A.J. Schmidt, K.C. Collins, A.J. Minnich, and G. Chen: J. Appl. Phys., 2010, vol. 107, pp. 104907-1–5. 37. A.J. Schmidt, K.C. Collins, A.J. Minnich, and G. Chen: Rev. Sci. Instrum., 2008, vol. 79, pp. 114902-1–9. 38. J.C. Duda, J.L. Smoyer, P.M. Norris, and P.E. Hopkins: Appl. Phys. Lett., 2009, vol. 95, pp. 031912-1–3.
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METALLURGICAL AND MATERIALS TRANSACTIONS B
K. JAGANNADHAM Samples of graphene composites with matrix of copper were prepared by electrochemical codeposition from CuSO4 solution with graphene oxide suspension. The thermal conductivity of the composite samples with different thickness and that of electrodeposited copper was determined by the three-omega method. Copper-graphene composite films with thickness greater than 200 lm showed an improvement in thermal conductivity over that of electrolytic copper from 380 W/m.K to 460 W/m.K at 300 K (27 °C). The thermal conductivity of coppergraphene 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 the composite samples and determine the interfacial thermal conductance between copper and graphene. The values of interface thermal conductance greater than 1.2 GW/m2.K obtained from the acoustic and the diffuse mismatch models and from the EMA modeling of the experimental results indicate that the interface thermal resistance is not a limiting factor to improve the thermal conductivity of the copper-graphene composites. DOI: 10.1007/s11663-011-9597-z Ó The Minerals, Metals & Materials Society and ASM International 2011
I.
INTRODUCTION
LOW-COST manufacturing of thermal interface materials and heat spreaders with high capacity to dissipate thermal energy is of significant importance in nanoelectronics and high-frequency and high-power devices 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 higher compared with other materials. Of these, the measurements at room temperature showed that the thermal conductivity of free-standing graphene is the highest with the value between 3000 and 5000 W/m.K and, thus, greater than that of other forms of carbon discovered until now. However, it was also shown recently[9] that the thermal conductivity of monolayer of graphene supported on SiO2 is only 600 W/m.K as a result of phonon leaking and phonon scattering by the substrate. To alleviate the problems associated with low thermal conductivity thermal interface materials[10] (TIM), recently we studied a simple method of processing In-graphene and In-Ga-graphene composites[11,12] and found that the thermal conductivity at 300 K (27 °C) is improved by a factor of 2.5 and 3, respectively, compared with that of In and In-Ga. In general, a heat spreader system contains a high capacity heat spreader and an external cooling system in addition to the TIM attached to the active device region. In the current work,
K. JAGANNADHAM, Associate Professor, is with the Department of 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.
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we studied the preparation of copper-graphene (Cu-gr) composite heat spreaders by electrochemical codeposition from CuSO4 solution with graphene oxide suspension. The thermal conductivity of the Cu-gr composites was determined experimentally and modeled to determine the importance of interface thermal conductance and graphene particle size.
II.
EXPERIMENTAL PROCEDURES
The preparation of copper-graphene composite samples has been described at length in the previous work.[11,13] A brief description is provided here. Graphene oxide (GO) films were prepared by chemical exfoliation from microcrystalline graphite[14] supplied by Asbury Graphite Inc. (Asbury, NJ) The details of preparation of exfoliated GO are discussed at length in our previous work.[11] A suspension of GO particulates in a 0.2 M solution of CuSO4 was prepared. Electrochemical codeposition from a bath containing graphene oxide (GO) suspension in a solution of technical grade CuSO4 in distilled water was carried out on oxygen-free high-conductivity (OFHC) copper foils. The pH was maintained close to 7, and a low current density of 1.75 mA/cm2 and growth rate of 2 to 3 lm/h[13] were used with a pure Cu anode to achieve smooth films. GO is hydrophobic on the basal plane and hydrophilic at the edges so that the suspension of the GO particulates is maintained 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 to smaller size of approximately 8-mm 9 10-mm area using a diamond saw. To evaluate the improvement in
METALLURGICAL AND MATERIALS TRANSACTIONS B
the thermal conductivity introduced by graphene, samples of electrolytic Cu were also deposited on the Cu foil from the same electrolytic bath but without GO suspension. The thickness of OFHC Cu foil in all the samples was 135 lm, but the thickness of the deposited composite film on the opposite sides of the Cu foil varied. The electrolytic Cu and Cu-gr samples were heated in flowing hydrogen atmosphere at partial pressure of 20 Torr and 673 K (400 °C) for 3 hours to reduce GO to graphene and Cu2O or CuO to Cu and thus form Cu-gr composites. Additional characterization by scanning electron microscopy (SEM) for graphene morphology and distribution of graphene platelets, energy dispersive spectrometry (EDS) for presence of oxygen and other impurities, and electrical conductivity as a function of temperature were carried out.[13] The electrical resistivity and temperature coefficient of resistance (TCR) were used previously to determine the volume fraction and resistivity of thinner films of graphene in all the samples.[13] SEM imaging combined with lineal fraction analysis in quantitative metallography[16] was used to determine the volume fraction of thicker graphene in the Cu-gr samples. Thermal conductivity of the samples was measured using the 3-x method. The details of the 3-x method are described at length in the previous work[12] and in the original Reference 17. A schematic diagram illustrating the sample with a gold heater line on the surface is presented in Figure 1. In the 3-x method, a line heater is used both as a source of heat and to measure the temperature on the surface of the sample for which the thermal conductivity is measured.[12,17] The Cu-gr composite sample surface is isolated by spreading an electrically insulating polymer film. The polymer film is a commercial thermosetting white epoxy S-30/3045 supplied by Devcon (Ellsworth Adhesives, Germantown, WI). In addition, thin films of insulating Si and ZrO2 each with thickness approximately 0.2 lm were deposited by laser physical vapor deposition (LPVD) on the top of the polymer film to enable the adhesion of a gold heater line to the surface. The gold heater line, with a width below 200 lm and length up to 8 mm, was deposited by LPVD using a negative stainless steel mask placed on the surface. Indium foils, with a width of 1 mm and length of 5 mm, were pressed on to either end of the gold heater line to make electrical contacts using
gold wires of 0.1 mm diameter. Two contacts on the opposite sides of the heater line were used to input the current and the other two were used to measure the output of voltage. The samples were mounted using ZnO paste on the substrate stage of a temperature-controlled vacuum chamber supplied by MMR Technology (Mountain View, CA). The chamber was kept under vacuum at 6 millitorr using a mechanical pump during electrical measurements. The temperature of the sample stage was controlled to ± 0.1 K by K-20 controller in conjunction with high-pressure nitrogen gas that was passed through a nozzle in the stage system of the chamber or by the Joule-Thomson effect. The measurements were made at three 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 was reached, the internal source of the lock in amplifier was used to supply power at the set input voltage V1 and frequency f, and the output voltage V3 was measured at frequency 3f. SR 830 DSP lock in amplifier supplied by Stanford Research Systems (Sunnyvale, CA) was used to supply the power at frequency f and measure the voltage at frequency 3f. The measurements were made as a function of frequency f (2pf = x) that was increased from 1 Hz to 3.5 kHz in different steps to cover the complete range. The electrical resistance (R) and temperature coefficient of resistance (a = dR/RdT) of the heater line were measured from the four contacts, with the two outer contacts to supply current at a constant value and the other two to measure the voltage. Keithley source meter 2400-LV and the nanovoltmeter 2182 (Keithley, Cleveland, OH) were used with reverse current procedure to cancel out the thermal electromotive force at the contacts. The average of the absolute value of the voltage was used to determine the resistance using Ohm’s law. The temperature was varied from 290 K to 315 K (17 °C to 42 °C) in steps 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 =ðaV1 Þ ½1
Fig. 1—Schematic illustration of the sample for the 3-x measurements. The sample with electrically insulating films on the top and the gold heater with indium foils attached on either side are shown. The gold wire electrical contacts on either side of the heater are also shown.
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where V1 is the voltage at frequency f, V3 at frequency 3f, and a is the temperature coefficient of resistance of the heater line. The power input per unit length into the heater was determined from V12/(lR) where l is the length of the heater line. A graph of dT per unit power input was generated as a function of ln(f) over the complete range of frequency at a given substrate temperature. These results were curve fitted using a multilayer analysis[18] with the help of a computer program. The parameters in the multilayer analysis consist 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. Interfacial thermal resistance between layers was also included in the multilayer analysis by introducing a layer with small thickness, heat capacity, and finite thermal conductivity. The multilayer analysis was started using the known values of thermal conductivity and heat capacity of
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different layers and the thickness of different layers shown schematically in Figure 2. The values of the parameters were modified during the numerical analysis to fit the experimental data and thus determine the thermal conductivity of Cu-gr layers. The polymer film had low thermal conductivity. The thermal waves penetrated into the ZnO paste only when the polymer film was thin. Therefore, an additional set of measurements were also made on two samples with thinner insulating polymer film. The thicknesses of the polymer film and the insulating layer are given in the results of curve fitting described subsequently.
III.
EXPERIMENTAL RESULTS
A. Results of Characterization of the Cu-gr Composite Films An SEM image of the cross section of the composite sample is provided in Figure 3 along with the energy dispersive spectrometry of the graphene film in Cu in Figure 4. The SEM image in backscattering mode and the X-ray maps of the carbon and the Cu signals are shown in Figure 3. The labeling in Figure 3 shows the pure Cu foil in the center and the Cu-gr composite films outside. The image also contains some graphene particulates that were sticking on the surface of the pure Cu film in the center as a result of contamination from cutting with diamond saw. It is observed from this image that graphene is distributed uniformly in the composite films outside the Cu foil. The X-ray map of carbon signal is shown in the middle of Figure 3.
The carbon signal in the center region of Cu is to the result of instrumental noise. If the noise from the signal is taken into account, then the middle region of Cu has no carbon but the outer regions of Cu-gr have a higher carbon signal. The Cu signal shown on the right-hand side of Figure 3 contains strong Cu signal from the center and a reduced Cu signal from Cu-gr composite. The EDS spectrum obtained from graphene particulates in the Cu-gr film is shown in Figure 4. The spectrum contains strong C and Cu peaks and a small O peak. The oxygen peak is associated with residual concentration in the film. X-ray diffraction[13] from a large area of the Cu-gr films in the as-deposited condition showed a diffraction peak at 2h = 11.8 deg that is associated with graphene oxide particulates but did not show any peak associated with graphite at 2h = 26.5 deg. This result indicated that the composite samples did not contain any graphite. However, the GO peak was not present on annealing in hydrogen at 673 K (400 °C) for 3 hours, which indicated that GO is converted to graphene layers of small thickness. Transmision electron microscopy[13] of the suspended GO particulates on microscopy grid also showed thin films with a diffraction pattern containing diffraction spots arranged in a hexagonal pattern. These results presented in the previous work[13] and the SEM characterization
Fig. 2—Schematic illustration of the different layers in the cross section of the composite samples prepared for thermal conductivity measurement. The gold heater wire and indium contacts are not shown.
Fig. 4—EDS spectrum of graphene particulates on Cu matrix in Cu-gr composite films. The residual oxygen peak is observed in addition to the major carbon and Cu peaks.
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 the left. 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 on the 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 the Cu foil region in the center were present because of contamination during cutting with the diamond wheel.
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confirmed that the composite films contained graphene platelets in a Cu matrix. B. Results of Thermal Conductivity A list of all the samples and the volume fractions of graphene evaluated from the measurement of electrical resistivity[13] and quantitative metallography are provided in Table I. First, the results obtained on the thermal conductivity of two samples of Cu-gr4 using a thinner and a thicker polymer insulating film on the top are described. Next, the results obtained on all the samples are presented along with the temperature dependence. The results obtained at 300 K (27 °C) from sample Cu-gr4-1, which consisted of Cu-gr composite films of thickness 240 lm deposited on the top and 205 lm at the bottom of the Cu foil are shown in Figure 5. The thermal conductivity, heat capacity, and thickness of the different layers used in the multilayer analysis are listed in Table II. The heat capacity of Cu and ZnO are taken from Reference 19. The thickness of the polymer film was 27.5 lm and that of the insulating ZrO2, Si, and the polymer was 11.8 lm. Thus, the total thickness of the top insulating films was 39.3 lm. An interfacial layer of 0.01 lm was used to represent the thermal resistance between the gold heater line and the top of ZrO2 film. The thermal waves penetrated to the bottom ZnO layer used for mounting the sample on the substrate holder in the chamber. The region of the graph with smaller slope in Figure 5 with 0 < ln(f) < 5 represents the bottom layers consisting of Cu-gr, Cu, Cu-gr, and ZnO. The region with the higher slope with 5 <ln(f) < 8.2 represents the top layers of electrically insulating films. The results obtained at 300 K (27 °C) from sample Cu-gr4-2, which consisted of Cu-gr composite film of thickness 215 lm deposited on the top and 165 lm at the bottom of the Cu foil, are shown in Figure 6. The thermal conductivity, heat capacity and thickness of the different layers used in the multilayer analysis are listed in Table III. The thickness of the polymer film was 66.1 lm and that of the insulating ZrO2, Si, and polymer layer was 13.2 lm. Thus, the total thickness
of the top insulating films was 79.3 lm, which is larger than that in the sample Cu-gr4-1. As stated, an interfacial layer of thickness 0.01 lm was used to represent the thermal resistance between the gold heater line and the ZrO2 film. The thermal waves have not
Fig. 5—Blue diamond symbols represent the experimental values of dT per unit power input and red squares represent the results obtained from multilayer analysis for sample Cu-gr4-1 with thinner polymer 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 the Experimental 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* Thermal Heat Conductivity Capacity Thickness (102 W/m.K) (106 J/m3.K) (lm) 0.0010 0.0060 0.0503 4.6 3.9 4.6 1.0 0.001 2.0 2.0 3.5 3.5 3.5 2.96 0.01 11.8 27.5 240 135 205 1200
Identified Layer Interface layer ZrO2, Si and polymer Polymer layer Cu-gr Composite Cu foil Cu-gr Composite ZnO
*The results of curve fitting are shown in Fig. 5. The polymer film plus the insulating layer on the top is relatively thin (39.3 lm). The temperature of the substrate was 300 K (27 °C).
Table I.
List of Samples Tested with Values of the Volume Fraction of Graphene fg Evaluated from Electrical Resistivity and Metallography* fg Thermal Conductivity (102 W/m.K) Metallography 0 0 0.25 0.24 0.26 0.19 0.20 Total Thickness (lm) 135 425 250 445 395 385 432 T = 250 K (–23 °C) 4.2 4.0 — 5.1 — — T = 300 K (27 °C) 3.9 3.8 4.5 4.6 4.6 4.6 4.6 T = 350 K (77 °C) 3.8 3.7 — 4.4 — — —
Sample Cu foil Cu-elec Cu-gr3 Cu-gr4 Cu-gr5 Cu-gr6 Cu-gr7
Resistivity 0 0 0.08 0.09 0.11 0.11 0.09
*The values of thermal conductivity of OFHC Cu foil, electrolytic Cu (Cu-elec), and Cu-gr composite samples determined by the 3-x method and analyzed by multilayer analysis are shown. The values of thermal conductivity are accurate to ±0.1. Cu-elec is electrolytic Cu Deposited on OFHC Cu foil of thickness 135 lm.
METALLURGICAL AND MATERIALS TRANSACTIONS B
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Fig. 6—The blue diamond symbols represent the experimental values of dT per unit power input. The red circles represent the results obtained from multilayer analysis for sample Cu-gr4-2 with thicker polymer film. The temperature of the substrate was 300 K (27 °C). Table III shows the different parameters used in the multilayer analysis.
Table III. Values of the Different Parameters used to Fit the Experimental 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* Thermal Heat Conductivity Capacity Thickness (102 W/m.K) (106 J/m3.K) (lm) 0.0010 0.00295 0.0070 4.6 3.9 4.6 0.001 2.0 2.0 3.5 3.5 3.5 0.01 13.2 66.1 215 135 165
film provided on the surface. Different parameters used in the analysis and the curve-fitted graphs are not presented in this article for the sake of brevity. The results were found to be the same from both samples, and the 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 the results are shown in Table I. The thermal conductivity of Cu-gr film in sample Cu-gr3 was found to be slightly lower than that of other samples, and this result is attributed to slightly variable microporosity introduced during electrochemical deposition. The temperature dependence of the thermal conductivity of the samples Cu-gr4-1 and Cu-elec with thinner polymer-insulating film on the surface was determined by 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 multilayer analysis. The values of the different parameters used in the multilayer analysis and the curve fitted graphs are not presented for the sake of brevity. The values of the thermal conductivity obtained for the Cu-gr films are shown in Table I.
Identified Layer Interface layer ZrO2, Si and Polymer Polymer layer Cu-Gr Composite Cu foil Cu-Gr Composite
IV.
DISCUSSION
*The results of curve fitting are shown in Fig. 6. The polymer film plus the insulating layer on the top is relatively thick (79.3 lm). The temperature of the substrate was 300 K (27 °C).
penetrated into the ZnO layer used for mounting the sample on the substrate holder because the polymer layer with lower thermal conductivity has lower thermal diffusion depth. The region of the graph with smaller slope in Figure 6 with 0 < ln(f) < 1 represents the bottom layers consisting of Cu-gr, Cu, and Cu-gr. The region with higher slope with 1 < ln(f) < 8.2 represents the top insulating layers. The multilayer analysis of the Cu-gr4-1 and Cu-gr4-2 samples with a different thickness of polymer film was carried out so that the larger region of smaller slope, as shown in Figure 5, could be curve fitted with more confidence. However, the values of the thermal conductivity of the Cu-gr layers shown in Tables II and III are same indicating that the analysis in Figure 6 is equally valid. The thickness of the Cu-gr composite film in both these samples is large (>200 lm), so that the thickness dependence of thermal conductivity did not change the value of thermal conductivity obtained in these two samples significantly. A multilayer analysis was also performed on the 3-x experimental data obtained from two samples of electrolytic copper (Cu-elec) film deposited on the Cu foil but with thinner and thicker electrically insulating polymer
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The results presented in Table I show that the thermal conductivity 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 Cu foil on which it is deposited. The slightly lower value is associated with a higher electrical resistivity[13] that is also observed in the electrodeposited Cu. An EDS analysis[13] of the samples showed presence of higher oxygen level in all the electrodeposited samples, and it is attributed to the impurities in the technical grade CuSO4 used in the electrochemical deposition. The details of the impurities and the higher electrical resistivity observed in the electrodeposited films are presented in the previous work.[13] Therefore, a subsequent comparison of thermal conductivity of Cu-gr films will be made with respect to the electrodeposited Cu film instead of the OFHC 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 smaller value of 4.5 W/cm.K in the sample Cu-gr3 with smaller total thickness of the composite film, as shown in Table I. This result indicates a large improvement from 3.8 W/cm.K associated with electrolytic Cu. The temperature dependence of thermal conductivity of the Cu-gr film in the sample Cu-gr4-1 is similar to that of electrolytic Cu in the temperature range of 250 K to 350 K (–23 °C to 77 °C). It is lower at higher temperature[20,21] as a result of increased scattering of electrons in Cu and phonons in graphene. These results are analyzed using effective mean field approximation (EMA). In particular, EMA is used to determine the dependence of thermal conductivity on the volume fraction of graphene incorporated in the Cu-gr composite films and to evaluate the interface thermal conductance between Cu and graphene.
METALLURGICAL AND MATERIALS TRANSACTIONS B
A. Evaluation of Volume Fraction of Graphene The volume fraction of graphene determined from a lineal fraction analysis using quantitative metallography[16] is presented in Table I. SEM imaging of graphene in backscattering mode showed only those graphene platelets that are thicker exhibit dark contrast. As a result, thinner graphene platelets such as single or bilayer graphene that exhibit weak contrast cannot be identified. Therefore, the volume fraction estimated from quantitative metallography is slightly lower than the actual value because the graphene films of smaller thickness are not included. The volume fraction of graphene determined from electrical resistivity and temperature coefficient of electrical resistance (TCR)[13] is also presented in Table I. The two unknowns, resistivity and the volume fraction of graphene, were solved using resistivity and the TCR of the composite samples in terms of the formulations of EMA.[13] Although this method is helpful to evaluate the volume fraction of thinner graphene platelets that contribute significantly to electrical conductivity, thicker graphene platelets that do not contribute to improvement cannot be evaluated. The electrical conductivity of graphene platelets thicker than 5 to 10 atomic layers approaches the characteristics of graphite that is semimetallic with lower electrical conductivity. Therefore, thicker graphene platelets do not contribute to electrical conductivity. However, both thinner and thicker graphene platelets contribute significantly to thermal conductivity[22–24] because thermal conductivity in the basal plane of graphene platelet is much higher than that of Cu. Therefore, the volume fraction of graphene that should be included in the analysis of thermal conductivity of Cu-gr composites is higher than the value obtained from quantitative metallography. The volume fraction of thinner and thicker graphene platelets in the sample is determined by adding the values obtained from electrical resistivity and quantitative metallography, respectively. The volume fraction of graphene determined by electrical conductivity is much smaller and within the accuracy achieved by quantitative metallography, and therefore, the total volume fraction is 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 thermal conductivity of indium-graphene composites and carbon nanotube composites.[12,25] Therefore, the formulations will be described only briefly. Under this approximation, the isotropic thermal conductivity of the Cu-gr composite medium is given by È À Á É Kc ¼ Kg Km 3 À 2fg þ 2fg Kg È À ÁÉ ½2 = fg Km þ Kg 3 À fg þ afg where fg is the volume fraction of graphene and K stands for thermal conductivity, with subscript c for the composite (Cu-gr), g for graphene, and m for matrix or Cu. The parameter a is given by Km/hr where h is the interfacial thermal conductance between the matrix and
METALLURGICAL AND MATERIALS TRANSACTIONS B
graphene and r is the effective radius of the graphene platelets. In the current analysis, r is taken as half the average size of graphene platelets present in the composite samples. EMA is used to determine the interfacial thermal conductance, h from Eq. [2]. The experimental determinations[8,22] of Kg have shown that its value is high when the size of the graphene platelets is bigger, such as 5 lm. A modeling analysis[21,23] also indicates that the value is expected to be in the experimentally observed range. Modeling[24] showed also that the value of Kg decreases to 1000 W/m.K when the size of the graphene platelet is 1 lm. The size (L) dependence of thermal conductivity in graphene nanoribbons has been shown[25] to follow (L)b, where b remains between 0.35 and 0.45. In addition, Klemens pointed out[26,27] that thermal energy leaks in to the matrix when graphene is present on a substrate, thereby, the thermal conductivity will be reduced further by 20 to 50 pct if the phonon velocity in the matrix is smaller than that in graphene. This result has been confirmed recently from the thermal conductivity measurements of graphene suspended on SiO2.[9] The average size Lav of the graphene platelets and the weighted average of the thermal conductivity Kg were determined from the distribution of the platelet sizes in the graphene suspension.[12] The same graphene suspension is used in the current experiments so that the average size of graphene platelets is 0.64 lm and the average value of Kg is 800 W/m.K. EMA is performed with these values of the parameters and in particular a = Km/hr where r = 0.32 lm. The results of interfacial thermal conductance h obtained from Eq. [2] are shown in Table IV. The results shown in Table IV illustrate that the interface thermal conductance remains between 1.2 and 2.2 in units of 109 W/m2.K, which is reasonably high. The variation in h between the samples is caused by the difficulty to determine the volume fraction of graphene accurately in the composites. C. Interface Thermal Conductance from Acoustic and Diffuse Mismatch Models The thermal conductivity in Cu is electron mediated, whereas that in graphene is phonon mediated in the temperature range of 250 K to 350 K (–23 °C to 77 °C).
Table IV. Value of Interfacial Thermal Conductance h Determined 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* Composite Sample Cu-gr3 Cu-gr4 Cu-gr5 Cu-gr6 Cu-gr7 Kc (102 W/m.K) 4.5 4.6 4.6 4.6 4.6 fg 0.33 0.33 0.37 0.30 0.29 h (109 W/m2.K) 1.2 1.5 1.3 1.9 2.2
*fg in this table is the sum of the two values of the volume fraction shown in Table I. See the explanation in Section IV.
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Table V. Parameters used in the Calculation of the Interface Thermal Conductance* Velocity (103 m/s) Medium Cu Graphene Density (106 gm/m3) 8.93 2.26 Longitudinal vl 4.8 24.0 Transverse vt 2.4 16.0 Debye vd 2.8 18.6 Lattice Specific Heat (106 J/m3.K) 3.5 1.84 c (103 J/m3.K2) 0.097 0.008 vf (106 m/s) 1.57 1.10
*Cu is treated as three-dimensional medium (3/v2 = 1/v2 + 2/v2) and graphene as two-dimensional medium (2/v2 = 1/v2 + 1/v2) for calculating d l t d l t the 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. The values of c and the Fermi velocity vf are taken from Refs. 29 and 30 for Cu and from Refs. 31 and 32 for graphene.
The interface thermal conductance is evaluated previously[12] using two approaches. In the first, we consider electron–phonon coupling in Cu and phonon–phonon coupling between Cu and graphene. Next, electron– electron coupling between Cu and graphene and electron–phonon coupling in graphene are used in the second approach. The electron–phonon coupling term hep in Cu is determined using[28] À Á1=2 Gep ¼ Cem =s and hep ¼ Gep Kpm ½3 where Cem is the electronic specific heat of Cu and s is the relaxation time for electron–phonon energy transfer. The values of different parameters used in the modeling are provided in Table V.[29–32] The electronic specific heat is calculated using the Cem = cT where c = 97.0 J/m3.K from Table V.[29] The relaxation time is assumed to be 1 ps[28] so that Gep = 29.1 9 106 GW/m3 at 300 K (27 °C). The phonon thermal conductivity of Cu is calculated from Kpm = Clmvdmlp/3 where Clm is the lattice 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 conductance from electron–phonon coupling hep is 6.87 9 108 W/m2.K. The phonon–phonon contribution hpp to the interface thermal conductance is determined from the acoustic or the diffuse mismatch models.[33,34] In the acoustic mismatch model[33] hpp ¼ Clm vdm a=4 ½4 where a is the transmission coefficient determined from À Á2 ½5 a ¼ 4Zm Zg = Zm þ Zg where Z is the acoustic impedance of each medium given by the product of density and Debye velocity, and the subscripts m and g refer to Cu and graphene, respectively. The value of hpp from acoustic mismatch model is determined using the values of parameters listed in Table V and found to be 2.26 9 109 W/m2.K. The expression for hpp in the diffuse mismatch model[33] is n À o Á hpp ¼ 1:02 Â 1010 T3 R 1=v2 R ð1=v2 m g n o 2 2 = R ð1=vm þ 1=vg ½6 In Eq. [6], the summation is performed over the longitudinal and the two transverse modes in Cu but
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only over the longitudinal and the single transverse mode in graphene because it is a two-dimensional medium. Using the parameters given in Table V, the value of hpp is found to be 1.50 9 109 W/m2.K. The net thermal conductance[28] determined from 1/h = 1/hep + 1/hpp is 4.7 9 108 W/m2.K from the diffuse mismatch model (DMM) and 5.3 9 108 W/m2.K from the acoustic mismatch model (AMM). An alternative expression from diffuse mismatch model[34] is, À Á ½7 hpp ¼ Clm vdm Clg vdg =4 Clm vdm þ Clg vdg where C represents, as before, the lattice specific heat and vd the Debye velocity. Using the parameters provided in Table V, the value of hpp = 18.9 9 108 W/m2.K. In this case, the total thermal conductance is found to be 5.0 9 108 W/m2.K. From the preceding results, the values of h from the acoustic and the diffuse mismatch models are close. In the next representation, the total thermal conductance will consist of electron–electron coupling between Cu and graphene and electro–phonon coupling in graphene. The first part is determined from,[35] À Á hee ¼ Ceg vfg Cem vfm T=4 Ceg vfg þ Cem vfm ½8 where vfg represents the Fermi velocity in graphene and vfm in Cu, Ceg is the electronic specific heat of grapheme, and Cem is that of Cu. The electronic specific heat is determined as before from Ce = cT and the values of c are listed in Table V for each medium so that the value of hee at T = 300 K (27 °C) is found to be 18.6 9 1010 W/m2.K. The electron–phonon coupling term[28] in graphene hep is determined as before from hep = {Gep Kpg}1/2 where Gep = (Ceg/s) and Kpg is the phonon thermal conductivity in graphene. Using the parameters listed in Table V, Gep = 2.38 9 106 GW/m3.K. The phonon thermal conductivity Kpg in graphene is size dependent. The distribution of graphene platelet sizes in the suspension is evaluated, and using the size dependence[25] of thermal conductivity, the average value was shown previously[12] to be 800 W/m.K so that hep = 1.38 9 109 W/m2.K. The net thermal conductance h given by 1/h = 1/hee + 1/hep is found to be 1.37 9 109 W/m2.K. Therefore, the value of thermal conductance based on electron–electron coupling is higher than when phonon–phonon coupling is used between Cu and graphene. The different values obtained for the interface thermal conductance from the various components and the total
METALLURGICAL AND MATERIALS TRANSACTIONS B
thermal conductance using the two approaches of carrier coupling are listed in Table VI. Although the electron– phonon coupling term in graphene is weak, it is stronger than in Cu by a factor of two. Similarly, the DMM indicates that electron–electron coupling term is stronger than phonon–phonon coupling term between Cu and graphene. The value of the interface thermal conductance, from Table VI, is lower than the experimental value shown in Table IV when phonon–phonon coupling is considered between Cu and graphene. The value obtained from electron–electron coupling, as shown in Table VI, is closer to the experimentally determined value using EMA. The lower Debye velocity of sound in Cu compared with that in graphene by almost a factor of 7, as shown in Table V, indicates clearly that quenching of phonons in graphene will take place.[27] Therefore, the thermal conductivity Kg of single- and double-layer graphene platelets will be reduced both from quenching of phonons and smaller size of graphene platelets. From this point of view, graphene platelets of 4 to 10 atomic layers thick wherein the phonon transport within inner layers is still effective seem to be advantageous. It is important to determine the thickness dependence of thermal conductivity of graphene platelets, especially when in contacts with a matrix such as Cu. It is useful to compare the current results with that of interface thermal conductance between metals and graphite obtained experimentally[36,37] and using modeling.[38] Although, these studies did not examine directly the interface thermal conductance between Cu and graphite, the results obtained for Al, Au, Ti, and Cr on C-axis-oriented graphite are relevant. The results showed that the interface thermal conductance remained between 30 and 100 MW/m2.K with lower range values applicable to noninteracting face-centered cubic metals like Au and higher values for interacting metals like Ti and Cr. The larger value of 1.0 9 108 W/m2.K in these measurements and modeling using phonon coupling[36,37] is only smaller by a factor of 5 from the DMM approach used in the current work. The main discrepancy is associated with the C-axis orientation of graphite crystals on which these studies were performed. The thermal conductivity of a few layers of graphene is three orders of magnitude smaller in the C-direction compared with the value in the basal plane or normal to the C-direction. The values shown for the present work assume thermal transport in the basal plane of graphene platelets, and hence, the interface thermal conductance is also expected to be much higher.
Table VI.
The modified DMM approach with phonon coupling used in Reference 38 estimates the interface thermal conductance between basal plane graphite and Al at 300 K (27 °C) to be close to 4 9 108 W/m2.K. This value is close to that obtained in the current calculations using phonon–phonon coupling in the DMM approach. The thermal conductivity of Cu is a factor of 2 higher than that of Al and that of graphene is also higher than that of graphite so that the interface thermal conductance between Cu and graphene is expected to be larger than that between Al and graphite. More importantly, the thermal conductance using electron–electron coupling seems to be higher than that using phonon– phonon coupling between Cu and graphene. The higher experimental value of interface thermal conductance in the current analysis using EMA could arise from the high thermal conductivity of both Cu and graphene.
V.
CONCLUSIONS
The thermal conductivity of electrochemically deposited Cu-gr composite and copper films on OFHC copper foils has been determined by the 3x method and multilayer analysis. The value of thermal conductivity of electrochemical deposited Cu has been found to be 380 W/m.K at 300 K (27 °C) and thus lower than that of OFHC Cu because of the oxygen introduced from the impurities in the CuSO4 electrochemical deposition bath. In contrast, the thermal conductivity of Cu-gr composite films is increased to 460 W/m.K at 300 K (27 °C) by incorporation of graphene through suspension in the electrolytic bath. The thermal conductivity of Cu-gr films measured in the temperature range of 250 to 350 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). The volume fractions of graphene determined from quantitative metallography and electrical resistivity measurements were used to determine the volume fraction that is closer to the actual value. EMA is used to determine the interface thermal conductance between Cu and graphene from the average size of the graphene platelets and thermal conductivity of Cu and graphene. Electron–electron coupling between Cu and graphene using diffuse mismatch model and electron–phonon coupling within graphene is found to give higher value of the interface thermal conductance and the value was found to be closer to that
Different Components of Thermal Conductance Across Cu and Graphene Interface* Total Thermal Conductance h 5.3 4.7 5.0 Electron–Electron and Electon–Phonon Coupling hee (Cu- graphene) 1860 (DMM) hep (graphene) 13.8 Total Thermal Conductance h 13.7
Electron–Phonon and Phonon–Phonon Coupling hep (Cu) 6.9 6.9 6.9 hpp (Cu-graphene) 22.6 (AMM) 15.0 (DMM) 18.9 (DMM)
*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.
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determined from EMA. It is concluded that the interface thermal conductance, close to 1.4 GW/m2.K, is not the limiting factor to achieve a greater improvement in the thermal conductivity of the Cu-gr composites. Instead, the smaller size of graphene platelets with a resultant lower average value of thermal conductivity of graphene is thought to be the limiting factor. In addition, thicker graphene platelets with more than three atomic layers are expected to possess higher thermal conductivity in the presence of matrix so that the quenching of phonons that carry the heat in graphene is confined only to the outer layers. ACKNOWLEDGMENT This research is supported by National Science Foundation Grant CMMI #1049751.
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METALLURGICAL AND MATERIALS TRANSACTIONS B
