1. Introduction

The GaN materials system has established itself as being very important for the next generation of high-power density devices for optical, microwave, and radar applications Reference Balandin, Morozov, Cai, Li, Wang, Wijeratne and Viswanathan[1] Reference Balandin[2] Reference Balandin, Cai, Li, Wang, Rao and Viswanathan[3] Reference Wu, Keller, Keller, Xu, Thibeault, Denbaars and Mishra[4] Reference Eastman, Tilak, Smart, Green, Chumbes, Dimitrov, Kim, Ambacher, Weimann, Prunty, Murphy, Schaff and Shealy[5]. At the same time, performance of these devices has been limited by self-heating Reference Balandin, Morozov, Cai, Li, Wang, Wijeratne and Viswanathan[1] Reference Gaska, Osinsky, Yang and Shur[6]. Thus, accurate modeling of heat diffusion and self-heating effects in AlGaN/GaN heterostructures and device optimization based on such modeling become crucial for further development of nitride technology. Simulation of heat diffusion in GaN and related materials is complicated by large discrepancy in the reported experimental thermal conductivity data and its dependence on defects and dislocations Reference Kotchetkov, Zou, Balandin, Florescu and Pollak[7] Reference Zou, Kotchetkov, Balandin, Florescu and Pollak[8] Reference Daly, Maris, Nurmikko, Kuball and Han[9]. We have previously shown that the temperature rise in AlGaN/GaN heterostructure field-effect transistors (HFETs) is different for doped and undoped channel devices Reference Filippov and Balandin[10].

Recently, there have been experimental indications that the overall thermal resistance of AlGaN/GaN device structures is larger than the simple model estimates from the acoustic mismatch theory (AMT) Reference Eckhause, Süzer, Kurdak, Yun and Morkoç[11]. One of the possible explanations of this fact can be a relatively large thermal boundary resistance (TBR) at the interface between GaN layer and the substrate. It has been experimentally determined in Ref. Reference Eckhause, Süzer, Kurdak, Yun and Morkoç[11] that the TBR of the GaN/sapphire interface at 4.2K is about three orders of magnitude higher then AMT predictions. A strong effect of TBR on heat diffusion in device structures has been observed for other materials systems Reference Kim, Kwun, Lee, Seo and Yoon[12] Reference Goodson, Käding, Rösner and Zachai[13] Reference Hu, Kiene and Ho[14].

In this paper, we calculate TBR for GaN/SiC, GaN/sapphire and GaN/AlN interfaces using the diffuse mismatch model (DMM). The obtained values are then used to simulate heat diffusion and temperature rise in GaN/AlGaN HFETs with characteristic biasing parameters.

2. Thermal Boundary Resistance

TBR is used to describe thermal transport across an interface and is defined as the inverse of thermal boundary conductivity

Here

is a heat flow across an interface, Ais an area and ΔT is the temperature difference between the two sides of the interface. In order to calculate TBR at the interfaces between different layers in a HFET structure we use the DMM approach, which assumes that the phonons incident on the interface will all undergo diffuse scattering Reference Swartz and Pohl[15]. In the framework of this model, TBR can be written as

where

Here k_B is Boltzmann's constant, h is Planck's constant, and the averaged transmission coefficients are given by

where ν_i,j are phonon velocities, index i = 1 stands for GaN and i = 2 for SiC, sapphire or AlN. Another index j = 1,2,3 indicates longitudinal (LA) and two transverse (TA) sound velocities, respectively. The limit of integration is the Debye frequency

where

is the average sound velocity in GaN. The cut off wave vector is given by

where N_A is the Avogadro number, ρ is the mass density, M is the atomic weight.

Using the definition

we estimated Debye temperature for GaN to be θ^Debye ≈ 614K if LA and TA sound velocities values are taken along [0 0 1] and θ^Debye ≈ 716K if the sound velocities values are taken along [1 0 0]. The first value coincides with the one given in Ref. Reference Bougrov, Levinshtein, Rumyantsev and Zubrilov[16]. Taking numerically the integral in Equations 2 and 3 with the materials parameter values from Refs. Reference Swartz and Pohl[15] Reference Bougrov, Levinshtein, Rumyantsev and Zubrilov[16] Reference Goldberg, Levinshtein and Rumyantsev[17] Reference Goldberg, Levinshtein and Rumyantsev[18], we obtained TBR as a function of temperature. We have also evaluated TBR in the low temperature limit, where the integral of Equation 2 reduces to the Riemann Zeta function. The calculated TBR value for the GaN/sapphire interface at 4.2K is 0.31 10⁻⁴ m²K/W which is closer to the experimentally determined value of 1.05 10⁻⁴ m²K/W in Ref. Reference Eckhause, Süzer, Kurdak, Yun and Morkoç[11] than the estimates from AMT. The room temperature TBR values for GaN interface with relevant materials are summarized in Table 1. The temperature dependence of the GaN/SiC TBR is shown in Figure 1.

Figure 1. Thermal boundary resistance as a function temperature for GaN/SiC interface. Results are shown in semi-log scale. The dashed line corresponds to the low temperature approximation.

One should note here that based on the data reported for other materials systems the TBR values calculated from Equations 2 and 3 correspond to the lower bound limit. The actual TBR can be up to an order of magnitude larger depending on the interface quality and roughness Reference Hu, Kiene and Ho[14].

3. Heat Flow in AlGaN/GaN HFET

Using TBR values obtained in the previous section, we simulated heat diffusion in AlGaN/GaN HFET layered structure. To obtain the temperature rise in the device structure we numerically solve the nonlinear heat flow equation

Here u = T − T_a is the temperature rise above ambient temperature T_a=300K, k is thermal conductivity taken as in Refs. Reference Zou, Kotchetkov, Balandin, Florescu and Pollak[8], c is specific heat, ρ is mass density,

is the heat-source term defined for the fixed generated power P and thin region of the following dimensions: a = 0.01 μm L =1 μm (L =0.25 μm), and W = 200 μm. It is assumed that f=0 everywhere else, and that left, right, back, and front boundaries are far enough from the heat-generating region. The bottom of the substrate is maintained at the ambient temperature T_a = 300K via its good thermal contact (see Figure 2). The heat-generating region is positioned next to the gate contact, on the drain side, at the depth of 0.023 μm below the device top surface. The dimensions of the structures are chosen such as to allow comparison with experimental data in Ref. Reference Kuball, Hayes, Uren, Martin, Birbeck, Balmer and Hughes[19].

Figure 2. Layered structure of the AlGaN/GaN heterostructure field-effect transistor.

The boundary value problem set by Equation 7 has been solved numerically using the finite element method. The results of the simulation of temperature profiles for the GaN/AlGaN HFET on SiC substrate with the two different TBR values, assigned to the boundary elements, are shown in Figure 3. Comparing temperature profiles in Figure 3, one can see that 1.2 10⁻⁸ m²K/W value of the TBR leads to approximately 20% increase of the maximum temperature in the device channel. One can also note from the left panel in Figure 3 that even with the theoretically determined value of TBR the constant temperature curves undergo strong discontinuity at the GaN/SiC interface. For higher values of TBR the GaN/SiC interface acts as thermal insulator that keeps the heat in the active channel and creates a hot spot. Higher temperature may lead to mobility degradation and negative differential resistance. We have obtained the dependence of the maximum temperature on TBR for different substrates and heat source lengths (see Figure 4). For example, for SiC substrate with the GaN/SiC TBR value of 4 x 10⁻⁸ m²K/W the increase in the maximum temperature in the channel is up to 60%. As it can be seen from Figure 4, for the sapphire substrate, thermal boundary resistance is less significant than that for SiC substrate due to the relatively low value of thermal conductivity of sapphire.

Figure 3. Temperature profiles in GaN/AlGaN HFETs on SiC substrate for two different values of the thermal boundary resistance. Left panel shows the results for R = 1.2 10⁻⁹ m²K/W, right panel shows the results for R = 1.2 10⁻⁸ m²K/W. The dissipated power is P = 12 W/mm in both cases. Note the different temperature scale in two figures.

Figure 4. Temperature maximum in the drain-gate opening as function of the thermal boundary resistance for the GaN/SiC interface (dissipated power is P/W = 10W/mm) and for GaN/Sapphire interface (dissipated power is P/W = 2.5W/mm). The results are shown for the two different HFETs with L=250nmheat-source length (blue curves) and L=1mm (black curves).

The results of our heat flow calculations of the temperature rise for the AlGaN/GaN HFETs are in good agreement with experimental data of Ref. Reference Kuball, Hayes, Uren, Martin, Birbeck, Balmer and Hughes[19].

4. Conclusions

We theoretically investigated the thermal boundary resistance and heat diffusion in AlGaN/GaN heterostructure field-effect transistors on SiC substrate. From our calculations, based on the diffuse mismatch model, the room-temperature thermal boundary resistance at the interface between GaN and SiC is estimated to be 1.2 10⁻⁹ m²K/W. Solving the heat diffusion equation for a given device structure, we found that the GaN/SiC interface thermal boundary resistance can strongly influence the temperature rise in the AlGaN/GaN device channel.