FEM thermal analysis of Cu/diamond/Cu and diamond/SiC heat spreaders Open Access

Design of efficient thermal management system is highly important since packing and power density in integrated circuits increase faster than the cooling efficiency of substrate materials.¹ For proper functioning of the thermal management system in removing heat from the electronic devices, two main factors must be considered simultaneously: thermal conductivity and coefficient of thermal expansion (CTE) of the heat spreading/sinking material.² A mismatch in CTE can potentially harm the performance and mechanical stability of the system. Thus, the fabrications of highly thermal conductive material such as metal-matrix composites are exhaustively studied for this emerging demand.^3,4 Thermal conductivities above 20W/cmK and coefficients of thermal expansion of the order of 4-6×10^-61/K are used in conventional heat spreader materials.⁵ Due to its high thermal conductivity,⁶ chemical and mechanical stability,⁷ low dielectric constant and high dielectric strength,⁸ diamond films are one of the best candidates for thermal spreading device fabrication applications. Natural diamond has an extreme thermal conductivity of 20W/cmK, and is considered an ideal material for such applications. However, its extreme cost and scarcity handicaps the feasibility of fabricating heat spreader from natural diamonds. Fortunately, the invention of synthetic diamond using chemical vapor deposition (CVD), plasma assisted CVD, DC plasma jet systems and hot filament CVD techniques have circumvented the problem of cost and scarcity of natural diamonds. The utilization of diamond for thermal management application is mainly carried out in two ways: CVD coating and diamond-based metal matrix composites (MMCs) such as diamond/Cu, diamond/Al and diamond/Ag, etc,^5,9–13 Moreover, diamond–silicon carbide composites are also produced under high pressure, high temperature conditions.¹⁴ On the other hand, diamond/β-SiC composite films exhibiting a gradient composition were synthesized by a hot filament CVD technique utilizing H₂, CH₄, and tetramethylsilane as reaction gases.¹⁵ The effect of copper on the thermal management efficiency of diamond for heat sink applications is reported.¹⁶ Using CVD technology, the experimental fabrication and functionality of copper/diamond/copper heat spreader has been studied by Young et al.^17,18 FEM analysis is convenient for studying mechanical¹⁹ and thermal properties of diamond nanostructures.^20,21 Experimental verification of diamond–silicon carbide composite by sintering at high temperature and high pressure were demonstrated and surface stress distribution of the structure is discussed by Wieligor and Zerda.¹⁴ The performance of Cu/diamond/Cu heat spreader has also been reported.¹⁸ The effect of chromium carbide (Cr₇C₃) coatings on diamond particles for improving the wettability in the case of diamond reinforced copper matrix composite has been investigated.^22,23 The poor adhesion and high stress level for diamond or diamond like carbon on a metallic substrates is found to be dependent on the difference in coefficients of thermal expansion between the film and the substrate.^24–26 However, the study of such effect in the case of CVD coated diamond for thermal management applications is scarce. Thus, there is still a need to fill the gap by studying the effect of Cr₇C₃ on stress reduction that arises due to thermal coefficient mismatch between diamond and copper.

In this paper, we report the FEM simulation results of the thermal analysis of a Cu/diamond/Cu and diamond/SiC devices for thermal spreading. In Cu/diamond/Cu heat spreader a diamond thin film is embedded between Cu layers. The effect of carbide layer on interfacial stress reduction is analyzed using FEM software COMSOL Multiphysics. The details of the paper are presented in the subsequent sections as follows: The finite element method is presented and discussed in the second and third sections, respectively. Finally, the conclusions are made in the last section.

A common practice for reducing the local temperature microelectronic device packages such as high-power transistor devices is to insert a thin film of diamond material as a heat spreader between the device and the heat sink.²⁷ Dimensions of materials used in the simulation are shown in Table I and Table II. Due to its large thermal conductivity, the diamond layer can rapidly dissipate heat away from the heat source to the bottom Cu layer which is designed to connect to a heat sink. The forced convection at the bottom side is set to h=1500Wm⁻²K⁻¹ as if it has a large heat sink that can dissipate heat rapidly through forced air convection as suggested by Chen and Young.¹⁷ The heat flux thorough the outer surface of the bottom Cu layer depends on its surface temperature. Therefore, the environmental and the initial temperature are set to room temperature (25⁰C) and the rest of the surfaces are set to natural air convection of h=20 Wm⁻²K⁻¹. The physical properties of diamond such as thermal conductivity, Young’s modulus, and thermal expansion coefficients are given in Table III. Both 3D and 2D models are applied using FEM based on the dimensional and material properties summarized in Table I, Table II and Table III.

Fig. 1 shows copper/diamond/copper heat spreader thermal analysis illustrating the thermal performance of a 10mm by 10mm by 142μm sized heat spreader device. The top middle part is attached to an electronic component releasing heat at a temperature of T=127⁰C from the top side of the device with a convection coefficient of 1500 W/m²K on the other side. Fig. 1 shows that the surface temperature profile where the maximum and minimum temperature are 83.51°C and 58.7°C, respectively. The device structure is clearer from Fig. 5.

Fig. 2 shows the dependence of effective thermal conductivity of Cu/diamond/Cu heat spreader on diamond layer thickness. It shows that the whole device thermal conductivity increases with diamond thickness. To check the accuracy of the results, simulation results are compared with analytic predictions given by Ref. 31 and sown in Equation (1),

where $κi$ and d_i (i=1,2,3) are the thermal conductivities and thicknesses of materials 1, 2 and 3, respectively. The simulation and analytic results for thermal conductivity agree perfectly and the curves in Fig. 2 overlap. However, the thermal conductivity of the device decreases with diamond film thickness as shown in Fig. 3. This is due to the relatively lower thermal expansion coefficient $(α)$ of diamond (1.2×10⁻⁶/K) as compared to copper (17×10⁻⁶/K) or other materials tabulated in Table III.

The effect of diamond film thickness on the total heat flux of the Cu/diamond/Cu heat spreader device is shown in Fig. 4. The heat flux shown in Fig. 4 increases with both temperature and diamond film thickness. Thus, the advantage of using diamond in such device increases the heat flux in addition to improving mechanical strength (high Young’s modulus) and decreasing the thermal coefficient of the device as shown in Fig. 3.

Fig. 5 shows stress contour of diamond heat spreading device under a heat source at T=200⁰C without carbide interlayer. At the interfaces of copper/diamond the stresses are larger than elsewhere. In addition, the stress developed at the ends are larger than at the center along the interfacial lines. The large simulated stress values shown in Fig. 3 are attributed to thermal coefficient mismatch as tabulated in Table III.

While the integration of diamond with metals such as copper, aluminum, titanium, etc. is useful for design and construction of effective device for thermal management, still there exists challenge with handling of mismatch in thermal coefficient as one observes in Table III. A difference in thermal coefficient mismatch leads to dimensional mismatch³² interfacial stress and thermal fatigue³³ build up that may lead to failure in the device especially when operating at elevated temperatures.^24,34 A thermal expansion mismatch could also destroy the integrity of the bond between the IC and the heat spreader device as a result of thermal cycling during normal operation.³³ As a result of dimensional expansion and/or contraction that would normally occur in the device with variations in temperature, the magnitude of a thermal stress (σ) developed by a temperature change (⁠$ΔT$⁠) is dependent on the coefficient of thermal expansion (α) and the modulus of elasticity³³ (E) according to Equation (2). Due to film-substrate dimensional mismatch, strain energy accumulates in the growing film. An alternative way to tackle such problem is the use of metallic carbide interlayer at the interfaces of metal-diamond.^25,34 Fig. 6 shows the plot of maximum stress versus diamond layer thickness in Cu/diamond/Cu heat spreader device with carbide layer of thicknesses 1μm at the Cu-diamond interfaces. The maximum stresses correspond to interfacial stress shown in Fig. 5. The result compares stress values with/without Cr₇C₃ interlayer where the stress is reduced when carbide interlayer of 1μm thick is deposited at the Cu-diamond interfaces and shown in Table IV. The effect of film thickness on stress reduction presented in Fig. 6 agrees with the findings of Wei et al in maximum stress reduction with interlayer in diamond-like carbon.³⁴ Fig. 7 presents the effect of temperature dependence of maximum thermal stress devolved in the device with/without carbide interlayer. The maximum stress values increase with temperature while the addition of carbide interlayer reduces the stress. Moreover, due to the adhesion problem that may exist between diamond and copper, the addition of carbide interlayer can also improve the adhesion.²⁴ 

Whenever large thermal coefficient mismatch occurs between materials aimed for heat spreading, an alternative method is to use other constituent materials of comparable thermal coefficient. Diamond film grown on silicon carbide substrate is suggested and demonstrated for such applications. In the subsequent section, we will present the thermal response of diamond/SiC. The advantage of using diamond/SiC for heat spreading device emanates from the fact that the thermal expansion of diamond (1.2×10⁻⁶/K) is closer to that of SiC (4.3×10⁻⁶/K) than that of copper (17×10⁻⁶/K) and hence the device will experience less thermal stress.

Fig. 8 shows the effect of diamond film thickness on the heat flux of the diamond/SiC heat spreader device. The heat flux shown in Fig. 8 increases with temperature but it decreases with diamond film thickness. Thus, using thicker diamond film relieves stress in addition to enhancing the heat flux (as shown in Fig. 8) from the electronic device to a heat sink. In addition, the magnitude of thermally induced maximum stress shown Fig. 9 is lower compared to that of Cu/diamond/Cu heat spreader device show in Fig. 6. This is attributed to the pronounced thermal coefficient difference between copper and diamond as compared to SiC and diamond. Thus, in the later device one may not need an extra interlayer for better performance.

Fig. 10 shows the effective thermal conductivity of diamond coated silicon carbide heat spreader device. Analytic result from Equation (1) is used for comparison with simulation and experimental data (1.7 and 2.2W/cmK for 30 and 69μm, respectively) of Gray³¹ for the composite structure. In addition the simulation result is compared with analytic value from Equation (1). Similar procedure of thermo-mechanical characteristics of diamond heat spreader³⁵ and thermal stress reduction by introducing interlayer between film and substrate has been studied by Wei et al.³⁴ 

The interfacial thermal stress of a Cu/diamond/Cu heat-spreading device are analyzed. This work presents the simulated thermal stress and effective thermal conductivity with/without addition of carbide interlayer. The maximum thermal stress is observed at the interfaces of diamond and copper films. Adding a thin layer of chromium carbide at the interfaces can reduce the otherwise large interfacial stress resulting from large difference between thermal coefficient of copper and diamond. Similar model of diamond film grown on silicon carbide is simulated and compared. The interfacial stress in diamond/SiC interface reduces with diamond film thickness and has capable of performing at large temperature without experiencing destructive thermal failure.

This work was supported by the Science and Technology Coordinating Innovative Engineering Projects of Shaanxi Province (Grant No. 2014KTCQ01-33), Shaanxi, Xi’an, China.