.
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 [1] [2] [3] [4] [5]. At the same time, performance of these devices has been limited by self-heating [1] [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 [7] [8] [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 [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) [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. [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 [12] [13] [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.
TBR is used to describe thermal transport across an interface and is defined as the inverse of thermal boundary conductivity
|
.
| (1) |
Here
is a heat flow across an interface, A is an area and 
T
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 [15]. In the framework of this model, TBR can be
written as
 ,
| (2) |
where
 .
| (3) |
Here kB
is Boltzmann's constant, 
is Planck's constant, and the averaged transmission coefficients are
given by
 ,
| (4) |
where 
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
 ,
| (5) |
where
is the average sound velocity in GaN. The cut off wave vector is given by
 ,
| (6) |
where NA
is the Avogadro number, 
is the mass density, M
is the atomic weight.
Using the definition
we estimated Debye temperature for GaN to be 
Debye 
614K if
LA and TA sound velocities values are taken along [0 0 1] and Debye 716K
if the sound velocities values are taken along [1 0 0]. The first value
coincides with the one given in Ref. [16]. Taking numerically the integral in
Equations 2 and 3 with the materials parameter values from Refs. [15] [16] [17] [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. [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.
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 [14].
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
 | (7) |
Here u = T - Ta
is the temperature rise above ambient temperature Ta=300K,
k
is thermal conductivity taken as in Refs. [8], c
is specific heat,
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 Ta = 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. [19].
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.
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. [19].
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.
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