Thermal analysis of AlGaN/GaN HEMTs considering anisotropic and inhomogeneous thermal conductivity


 To examine the differences of thermal characteristics introduced by material thermal conductivity, anisotropic polycrystalline diamond (PCD) and GaN are analyzed based on the accurate model of grain sizes in the directions of parallel and vertical to the interface and an approximate solution of the phonon Boltzmann transport equation. Due to the space-variant grain structures of PCD, the inhomogeneous-anisotropic local thermal conductivity, homogeneous-anisotropic thermal conductivity averaged over the whole layer and the typical values of inhomogeneous-isotropic thermal conductivity are compared with/without anisotropic GaN thermal conductivity. The results show that the considerations of inhomogeneous-anisotropic PCD thermal conductivity and anisotropic GaN thermal conductivity are necessary for the accurate prediction of temperature rise in the GaN HEMT devices, and when ignoring both, the maximum temperature rise is undervalued by over 16 K for thermal boundary resistance (TBR) of 6.5 to 60 m2K/GW at power dissipation of 10 W/mm. Then the dependences of channel temperature on several parameters are discussed and the relations of thermal resistance with power dissipation are extracted at different base temperature. Compared with GaN, SiC and Si substrates, PCD is the most effective heat spreading layer though limited by the grain size at initial growth interface.


Introduction
Diamond composites is promising for thermal management of GaN high electron mobility transistors (HEMTs), due to its excellent heat spreading capability. Recent development of growth technology of polycrystalline diamond (PCD) facilitates the realization of large wafer sizes and a rather stable integration with GaN material. While the chemical vapor deposited PCD exhibits a columnar morphology with anisotropic grain size, which evolves with the distance from nucleation surface and is generally greater in the cross-plane direction than that in the in-plane direction. A. Sood et al [1] established a theoretical model of the PCD grain structures and related the anisotropy to phonon-grain boundary scattering, resulting in a rather practical expression of inhomogeneousanisotropic PCD thermal conductivity. Based on the phonon Boltzmann transport equation, Jungwan Cho et al [2] modeled the in-plane thermal conductivity of GaN and calculated its relation with layer thickness. Therefore, for accurate prediction of thermal transport in GaN power devices, it's necessary to consider the inhomogeneity and anisotropy in both GaN and PCD thermal conductivity.
Several thermal studies on GaN-on-diamond devices taking anisotropic and inhomogeneous PCD thermal conductivity into account, J. Anaya et al [3] analyzed the effect of the PCD grain structure on the phonon scattering and on its thermal conductivity, establishing an accurate quantification of the thermal management in a GaN high power amplifier with PCD substrate. B. Zou 3 et al [4] extracted an effective thermal conductivity of the PCD substrate as "seen" by the device via finite element thermal simulation, considering the depth-dependent anisotropic thermal conductivity of PCD. By additionally counting the thickness dependence of in-plane thermal conductivity of GaN, Song et al [5] estimated the fundamental limits for near-junction conduction cooling of high power GaN-on-diamond devices. In light of these investigations, our study models the space-variant thermal conductivity of PCD substrate as well as anisotropic thermal conductivity of GaN, compares the junction temperature differences in GaN HEMT brought by different thermal conductivity models in GaN and PCD, and explores various impacting parameters, TBR, GaN layer thickness, gate pitch, heat source width, base temperature, dissipated power and substrate materials on device temperature rise. Our research may provide some instructions on the accurate thermal design of GaN HEMTs on PCD substrate.

Simulation details
To analyze the thermal characteristics of AlGaN/GaN HEMTs, the schematic cross-section of the device structure is shown in Figure 1 (a), which consists of a GaN buffer layer, an 20 nm-thick interfacial layer at GaN/diamond interface incorporating effective interfacial thermal resistance, a PCD substrate layer, an AuSn solder layer and a CuMo heat sink [4]. The simulated transistor is comprised of 12 gate fingers. Figures 1 (b) and (c) display the zoomed-in view and top view of GaN layer. The overall length and width of the chip are L and W, respectively. The heat sources modeled as cuboids with an embedding depth of D e under GaN top surface is the representation of constant heat flux generated by dissipated power directly under the gate fingers, and gate/drain/source metallization is omitted due to the small structural complexity [4,6]. Here, a pitch spacing of S 4 between two adjacent heat sources is assumed. Due to the structural symmetry, only a quarter of the device is simulated, as illustrated in Figure 1(d). The structural parameters are summarized in Table   1 [4,6].   thermal conductivity data [2,7], yielding κGaN,In-plane=130 W/(mK) for layer thickness of 1.4 μm at room temperature. While for cross-plane thermal conductivity of GaN, a different heat flux suppression function that describes the reduction in phonon mean free paths due to boundary scattering is assumed based on the study by Zhang et al [8] and Sood et al [9]. Here, the parameter nv is fitted as 4.5×10 17 cm -3 with the reported data [10][11][12][13][14] and the resultant thermal conductivity at room temperature is ~160 W/(mK), in accord with the result in Ref. [5]. For simplicity, the correlation between GaN thermal conductivity and temperature is taken as kin-plane, cross-plane(T/300) -1.4 . in-plane and cross-plane thermal conductivity from literatures [2,7,[10][11][12][13][14].
As with PCD thermal conductivity, the key issue is to determine the average grain size at a certain distance from the growth interface based on phonon-grain boundary scattering. Following the model developed by A. Sood et al [1] that the columnar grains of PCD originating from the flat nucleation surface are trapezoidal regions, whose lateral sizes evolve linearly with the distance z from growth interface as and the vertical size can be deduced as [1]   where d0 indicates the grain size at the nucleation interface,  is the grain evolution rate, Ld is the thickness of diamond film, =   To simulate the thermal characteristics of GaN HEMTs, a constant heat flux generated by dissipated power is applied directly under heat sources, and the power dissipation is set to 10 W/mm.
The top AlGaN barrier layer is negligible in the analysis due to its minimal contribution on thermal resistance [4,6]. An isothermal surface of 300 K is applied at the bottom of the substrate and other external surfaces are assumed to convect naturally. Here, the natural air convection coefficient and ambient temperature are set as 20 W/(m 2 K) [19,20] and 300 K. For accuracy, the temperaturedependent thermal conductivity of the substrate layer is considered. The thermal parameters of each layer used in the simulation are summarized in Table 2. SiC 420(T/300) -1.3 581 3100 5 [13] AuSn 57 128 14700

Results and discussions
To assess the effect of PCD thermal conductivity on channel temperature of GaN-based HEMTs, we compare the cases with inhomogeneous and anisotropic thermal conductivity (red and blue bars in Figure 2(b)), homogeneous-anisotropic thermal conductivity (κr,avg and κz,avg mentioned above) and homogeneous-isotropic thermal conductivity (κr=κz=2000 W/(mK)). The results are displayed in For quantitative interpretation of the impact of inhomogeneous-anisotropic PCD and GaN thermal conductivity, we calculate the temperature differences introduced by neglecting neither the inhomogeneous-anisotropic PCD or anisotropic GaN, whose values are obtained by subtracting from the results with inhomogeneous-anisotropic PCD and anisotropic GaN thermal conductivity. As  Figure 4.
For thickness of 0.5~5 μm, the inhomogeneous-anisotropic PCD thermal conductivity contributes most to channel temperature, while homogeneous-isotropic and homogeneous-anisotropic PCD thermal conductivities affect almost equally. Similar to Refs. [21,25], an optimum GaN layer thickness exists due to the non-zero TBR and localized heating in active region of transistors, which is around 3 μm for TBR=10 m 2 K/GW and is about 4 μm for TBR=30 m 2 K/GW. In practice, for localized hotspot, the effect of GaN layer thickness on channel temperature is two-sided. On the one hand, the GaN layer needs to be thick enough to effectively conduct heat before through the highly resistive GaN/subtrate interface. On the other hand, a too thick GaN layer increases the thermal resistance due to the limited heat-spreading capability of GaN. Therefore, the optimum GaN layer thickness increases with the enlargement of TBR. The interrelations between the anisotropy and inhomogeneity in PCD thermal conductivity and gate pitch/heat source width are evaluated in Figure 5 (a) and (b), respectively. As seen, for gate pitch of 2~50 μm, the maximum temperature rise resulting from homogeneous-isotropic PCD thermal conductivity is almost the same with that from homogeneous-anisotropic PCD thermal conductivity 11 and is lower than that from inhomogeneous-anisotropic PCD thermal conductivity. The maximum temperature rise is large for gate pitch of 2 μm and drastically reduces by ~60% for inhomogeneousanisotropic PCD thermal conductivity and ~55% for homogeneous-isotropic/homogeneousanisotropic PCD thermal conductivity with gate pitch increasing to 10 μm. As gate pitch enlarges further, the temperature slowly decreases. Therefore, restraining the length of gate pitch is an effective way to reduce the channel temperature. Figure 5 (b) displays that the impact of heat source width is limited, since the increment of maximum temperature rise is only 10 K~15 K as heat source width increases from 50 to 300 μm.  Table 3.

Figure 6
Variation of maximum temperature rise with power dissipation at different base temperature for TBR of 30 m 2 K/GW.  [5,23,24,26], as summarized in Table 2. Except for the thermal conductivity of GaN homoepitaxy on GaN substrate, whose value is reported as high as 260 W/(mK) [27].The anisotropic SiC thermal conductivity is also introduced with ηSiC=κSiC, in-plane/κSiC, cross-plane=0.65 [13]. 13 As shown in Figure 7, the maximum temperature occurs at the proximity of active region in the transistor and is the largest in Si, then GaN, anisotropic SiC, isotropic SiC and PCD, which basically follows the orders of each thermal conductivities. Figure 8 (a) displays the power dissipation dependent temperature rises of inhomogeneous-anisotropic PCD with different TBR of 6.5, 30 60 m 2 K/GW, anisotropic/isotropic SiC, Si and GaN, among which Si substrate is the worst heat spreading layer, and the temperature rise can be reduced remarkably when using other substrate materials, especially for large power dissipation. The SiC substrate is superior to GaN substrate and the anisotropy in its thermal conductivity has minor effect on device junction temperature. PCD substrate proves its superior heat conducting capability even for large TBR value. The anisotropy ratios of the maximum temperature rise originating from isotropic versus anisotropic thermal conductivity in SiC and inhomogeneous-anisotropic versus homogeneous-isotropic thermal conductivity PCD are illustrated in figure 8 (b). The anisotropy ratio is evidently larger for SiC substrate than that for PCD substrate and becomes enlarged at higher power density due to more concentrated heat in the transistor.

Conclusion
The FEM heat simulation is used to estimate the effect of inhomogeneous-anisotropic PCD thermal conductivity and anisotropic GaN thermal conductivity on channel temperature. The results show that neglecting solely the anisotropy in GaN thermal conductivity underestimates the channel temperature by 7.3~15.5 K for TBR of 6.5~60 m 2 K/GW, while the differences for ignoring the anisotropic/space-variant PCD thermal conductivity are relatively large at small TBR, about 10 K at TBR of 6.5 m 2 K/GW and reduce to 6~7 K for TBR of 60 m 2 K/GW. For the case with homogeneous/isotropic PCD thermal conductivity and isotropic GaN thermal conductivity, the errors are the largest (over 15 K) at small TBR and display an increasing tendency when TBR increases.
Moreover, we analyze the dependences of channel temperature on several parameters, as GaN thickness, gate pitch, heat source width, base temperature, power and substrate material, and find that an optimum GaN layer thickness of ~3 μm exists for 0.5 μm-long gate at TBR=10 m 2 K/GW. 15 Restraining the length of gate pitch is an effective way to reduce the channel temperature, while the impact of narrowing heat source width is limited. The thermal resistance increases with increasing base temperature and power dissipation. Limited by in-plane thermal conductivity, PCD is still the most effective heat spreading layer among GaN, SiC and Si substrates.