Background

Gallium oxide (Ga2O3)-based metal–oxide–semiconductor field-effect transistors (MOSFETs) are excellent candidates for next generation power electronics, which are benefited from two major advantages of Ga2O3: the significantly high bandgap (~ 4.8 eV) and high-quality bulk crystals produced at low cost [1]. Tremendous efforts have been devoted to improving its electrical properties in all aspects like current density [2], breakdown voltage [3], and power figure-of-merit [4]. With the experimental confirmation of its unprecedented potential for power electronic devices [5,6,7,8,9], it is now of paramount importance to explore the performance and reliability of Ga2O3 MOSFETs, such as the issue of self-heating effects and hence the channel maximum temperature (Tmax), due to its relatively low thermal conductivity (κ, 0.11–0.27 Wcm−1 K−1 at room temperature) [1].

In recent years, various methods for estimating the Tmax of Ga2O3 MOSFETs have been proposed theoretically and experimentally [10,11,12,13]. In general, numerical simulations can quantitatively estimate Tmax of a certain device. However, this is time consuming [14]. On the other hand, the extraction of Tmax through experiments is always underestimated [15]. Therefore, an analytical model has to be made in order to adequately model the Tmax in Ga2O3 MOSFETs, which can provide sufficient accuracy with time-efficiency and qualitative assessments [14].

In this paper, we propose an analytical model of Tmax for Ga2O3 MOSFETs by employing Kirchhoff’s transformation, considering the dependence of κ on temperature and crystallographic directions for Ga2O3. The proposed model can be applied for Ga2O3 MOSFETs with native or high-thermal-conductivity substrates. The validity and the accuracy of the analytical model are methodically verified by comparison with the numerical simulations via COMSOL Multiphysics.

Methods and Model Development

The analytical model for Tmax in Ga2O3 MOSFETs is proposed based on the structure shown in Fig. 1. Key parameters of structure are listed in Table 1. In fact, it has been demonstrated that Joule heating is concentrated at the drain edge of the gate in Ga2O3 MOSFETs [13]. In order to simply the model, it is assumed that the heating effect from the gate is uniform [12] and can completely penetrate through the gate oxide due to its negligible thickness. Different substrate materials underneath Ga2O3 channel are considered in this model, such as bulk Ga2O3 and high κ materials, aiming at the board feasibility and compatibility. Thus, the device is viewed as a two-layer problem. The substrate contacts with an ideal heat sink so that the bottom surface is isothermal, and its temperature equals to that of ambient temperature (Tamb, 300 K by default). Adiabatic boundary conditions were imposed on other surface of the structure. These boundary conditions can be summarized as [14, 16]

Table 1 Key parameters of structure
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Fig. 1
figure 1

The schematic diagram of Ga2O3 MOSFET

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$${\kappa }_{y}{\left.\frac{\partial T}{\partial y}\right|}_{y={t}_{ch}+{t}_{sub}}=\left\{\begin{array}{c}\frac{P}{{L}_{g}} \left|x\right|\le \frac{{L}_{g}}{2}\\ 0 \left|x\right|>\frac{{L}_{g}}{2}\end{array}\right.,$$
(1)
$${\left.T\right|}_{y=0}={T}_{amb},$$
(2)
$${\left.\frac{\partial T}{\partial x}\right|}_{x=-\frac{L}{2}}={\left.\frac{\partial T}{\partial x}\right|}_{x=\frac{L}{2}}=0,$$
(3)

where P, T and κy denote the power dissipation density, temperature and thermal conductivity of [010] direction for Ga2O3, respectively. It should be emphasized that the unit of P is W/mm in this paper.

The κ value of Ga2O3, one of the key parameters for the thermal characteristic of material, plays an important role in the diffusion of heating effect as well as the accuracy of model. That is to say, a carefully description of κ value is required, due to its serious anisotropy and temperature-dependence [17]. In general, the dependence of κ of Ga2O3 on temperature (T) along two different crystal orientations ([001] and [010]) is given by

$${\kappa }_{\left[001\right]}\left(T\right)=0.137\times {\left(\frac{T}{300}\right)}^{-1.12},$$
(4)
$${\kappa }_{\left[010\right]}\left(T\right)=0.234\times {\left(\frac{T}{300}\right)}^{-1.27}.$$
(5)

The comparison study of Tmax at different P was carried out by COMSOL Multiphysics, considering constant and realistic κ, respectively. We found that at a P of 1 W/mm, Tmax values of 533 K and 622 K are obtained, respectively (not shown). Therefore, it is quite necessary to take into account the impacts of T and crystallographic direction on the κ of Ga2O3 in the model.

The temperature behavior is governed by the heat conduction equation. The heat conduction equation at steady-state in Ga2O3 domain is

$$\frac{\partial }{\partial x}\left({\kappa }_{x}\left(T\right)\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left({\kappa }_{y}\left(T\right)\frac{\partial T}{\partial y}\right)=0,$$
(6)

where κx denotes the thermal conductivity of [001] direction for Ga2O3. The nonlinear heat conduction equation can be solved by employing Kirchhoff’s transformation. However, the application of Kirchhoff’s transformation may be restricted due to the highly anisotropic κ in Ga2O3, which is valid, strictly speaking, only for materials with isotropic κ [14]. Given the above limitation, one should not consider κx and κy to be two independent variables. Figure 2 shows the relationship between the thermal resistivity, i.e., 1/κ, and T for directions of [001] and [010] over a large T range, respectively. It can be seen that 1/κy can be substituted with 1/(cκx) and c is chosen to be equal to 1.64. Consequently, Eq. (6) can be transformed to the following equation:

Fig. 2
figure 2

The relationship between the thermal resistivity and T for directions of [001] and [010]. Green symbols and red lines denote actual and fitted values, respectively. Blue line represents the hypothesis of 1/κy ≈ 1/(cκx), where c = 1.64

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$$\frac{\partial }{\partial \mathrm{x}}\left({\kappa }_{x}\left(T\right)\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial \mathrm{y}}\left({c\kappa }_{x}\left(T\right)\frac{\partial T}{\partial y}\right)=0.$$
(7)

Based on the preceding approximations of κx and κy, the Kirchhoff’s transformation can be employed without any restrictions. Besides, it also can be seen that the reciprocal of κ is expected to be proportional to T. Thus, in order to reduce the computational complexity, the expression of 1/κx can be simplified as 1/κx = aT + b, as shown in Fig. 2. The reason for the use of a, b and c is just convenience in writing the equations that follow.

By the application of Kirchhoff’s transformation and the method of separation of variables, the expression of Tmax can be derived as

$$\begin{aligned} T_{{max}} = & \\ & \,\left( {T_{{amb}} + \frac{b}{a}} \right)exp\left( {\frac{{aP\left( {t_{{ch}} + t_{{sub}} } \right)}}{{cL}} + \frac{{aPSL}}{{\sqrt c \pi ^{2} L_{g} }}} \right) - \frac{b}{a}, \\ \end{aligned}$$
(8)

where

$$S=\sum_{n=1}^{\infty }\frac{\mathrm{sin}n\pi \frac{{L}_{g}}{L}}{{n}^{2}}\frac{\mathrm{sinh}2n\pi \frac{{t}_{ch}+{t}_{sub}}{\sqrt{c}L}}{\mathrm{cosh}2n\pi \frac{{t}_{ch}+{t}_{sub}}{\sqrt{c}L}}.$$
(9)

It should be pointed out that S is a convergent infinite series and its approximate value which can be obtained easily is used in calculation instead of its actual value.

In the case of Ga2O3 MOSFETs with high κ substrates, Kirchhoff’s transformation cannot be directly applied theoretically. In fact, for the transformation to be valid, the boundary conditions should be either isothermal (constant T surface), or have a fixed heat flux density. However, due to the different κ of Ga2O3 and substrate material, both of these boundary conditions are not completely met at the Ga2O3/substrate interface. Considering that the κ of Ga2O3 is much lower than high κ substrate, a hypothesis, the isothermal interface between the Ga2O3 and the substrate, is introduced. This hypothesis is instrumental in deriving the expression Tmax and its validity will be verified later. In this case, the thermal resistance (RTH) of high κ substrate, a ratio of the difference between the Tint and Tamb and the PW, i.e., RTH = (Tint—Tamb) / (PW), can be calculated as RTH = LW/(κtsub), where W is the width of substrate [19]. Thus, the expression of the temperature of Ga2O3/substrate interface (Tint) is

$${T}_{int}=\frac{P{t}_{sub}}{{\kappa }_{sub}L}+{T}_{amb},$$
(10)

where κsub is the thermal conductivity of heterogeneous substrate, which is assumed to be constant. In addition, it should be pointed out that the thermal boundary resistance between Ga2O3 and heterogeneous substrates is not included in the model. Therefore, with the help of Eq. (8), the expression of Tmax for Ga2O3 MOSFETs with heterogeneous substrate can be derived as

$$\begin{aligned} T_{{max}} = & \\ & \;\left( {T_{{int}} + \frac{b}{a}} \right)exp\left( {\frac{{aPt_{{ch}} }}{{cL}} + \frac{{aPSL}}{{\sqrt c \pi ^{2} L_{g} }}} \right) - \frac{b}{a}, \\ \end{aligned}$$
(11)

where

$$S=\sum_{n=1}^{\infty }\frac{\mathrm{sin}n\pi \frac{{L}_{g}}{L}}{{n}^{2}}\frac{\mathrm{sinh}2n\pi \frac{{t}_{ch}}{\sqrt{c}L}}{\mathrm{cosh}2n\pi \frac{{t}_{ch}}{\sqrt{c}L}}.$$
(12)

Results and Discussion

The validity of the analytical model for the Tmax in Ga2O3 MOSFETs was systematically verified in this section, considering both native substrate and the counterpart with higher thermal conductivity. The best way to test the validity of a model is against experimental data. However, some key geometric parameters could not be found in experimental literatures, such as tsub and L in Ref. [12]. Therefore, finite-element simulation, one of the most accurate means, is used to verify our model. Figure 3 shows dependence of Tmax on power density P obtained from both COMSOL Multiphysics and analytical model, for Ga2O3 MOSFET with native substrate. Varied key parameters are considered, including device length L, substrate thickness tsub, and ambient temperature Tamb. As shown in Fig. 3a, the Tmax is naturally increased with the raised power density and the increase rate is boosted with the smaller L. This is attributed to that the device with larger L allows heat dissipation from the active region and hence its overall temperature is lower than that with smaller L at same P [11]. That is to say, its RTH, the slope of curves, is smaller than that of latter. Furthermore, since the κ of Ga2O3 will decrease with the increase in overall temperature, its RTH will also increase slower than that with smaller L consequently, which is obvious in Fig. 3a [19]. Similarly, the investigation of dependence of Tmax on tsub was performed, as illustrated in Fig. 3b. It is observed that the trend of Tmax with respect to P is same as that in Fig. 3a. The thinner substrate always produces the alleviated rise in Tmax over the enlarged power density, which is understandable that the thinner substrate, the lower overall temperature, the smaller RTH and its increase rate, just like the analysis in Fig. 3a. Figure 3c compares the influence of Tamb on Tmax as P increases. It is evident that the difference between two curves increases slowly, which is different from those in Fig. 3a, b. Ordinarily, RTH is dominated by the geometric parameters of device and the κ value of material. However, considering that the structure is fixed in this case, the increase in RTH is only induced by the decrease in κ of Ga2O3. On the other hand, a high level of agreement is observed for the proposed model, which considers the T- and direction-dependent relationship for the κ of Ga2O3, confirming the scalable nature of the model. On average, the difference of proposed model and simulation is < 1 K. The overall excellent agreement observed suggests that the proposed model is highly effective and accurate.

Fig. 3
figure 3

Dependence of Tmax on a the length of device L, b the thickness of substrate layer tsub, and c ambient temperature Tamb at different power P. Symbols and lines denote the results of proposed model and simulation, respectively

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Likewise, as shown in Fig. 4, the similar comparisons are repeated for Ga2O3 MOSFETs on high κ substrate, SiC. Here, the steps for L and Tamb that we choose are larger than those in Fig. 3, and the varied channel thickness tch is considered instead of tsub in this case. Otherwise, the difference between two curves of Tmax with respect to P in each figure will be undistinguishable, owing to the efficient heat dissipation capacity of SiC substrate. The κ of SiC (3.7 Wcm−1 K−1) applied is a default parameter in COMSOL Multiphysics software. Thanks to high κ of SiC, it can be seen clearly from all figures that the increase in Tmax is approximately linear as P increases, which means that the influence of temperature on the RTH of device is negligible. It should be pointed out that our model can describe this linear relationship successfully. However, it is obvious that the Tmax calculated by current model is lower than that predicted by simulation, and this difference is more evident with the increase in power consumption. To show this mechanism, simulated Tint are extracted with the power increasing and compared with calculated Tint by Eq. (10) as plotted in Fig. 5. It is found that the Joule heating becomes more concentrated in the middle of device as P increases. There are 0.5 K and 4 K ΔT between the model and simulation at this location when P = 0.25 and 1 W/mm, respectively. This is the reason that our model fails to accurately predict Tmax. Therefore, a more reasonable hypothesis of Tint is needed to obtain higher accuracy in future. Nevertheless, the Tmax is predicted by model to be only < 4 K lower than that by simulation even under 1 W/mm power dissipation density. That is to say, although the hypothesis of uniform Tint is inconsistent with fact, our model can provide an estimation of Tmax with enough accuracy.

Fig. 4
figure 4

Dependence of Tmax of Ga2O3 MOSFETs with SiC substrate on a the length of device L, b the thickness of Ga2O3 layer tch, and c ambient temperature Tamb at different power P. Symbols and lines denote the results of proposed model and simulation, respectively

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Fig. 5
figure 5

Comparison of Tint between simulated and calculated by Eq. (10) at different P

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Conclusions

An accurate analytical model to estimate the Tmax of Ga2O3 MOSFETs involving the temperature- and direction-dependent of thermal conductivity is presented. A simple expression based on device geometry and material parameters has been derived. An excellent agreement has been obtained between the model and COMSOL Multiphysics numerical simulations by varying different power consumption. The proposed model for the Tmax is of great importance for effective thermal management power devices especially Ga2O3 MOSFETs.