Improved Nonlinear Self-Heating Compact Modeling in SiGe HBTs Using Mextram
Authors: Zhang Anni , Niu Guofu , Li Yiao , Willemsen Marnix , Scholten Andries 
Published: December 2024

Accurate modeling of self-heating (SH) induced junction temperature rise is crucial for circuit design at high current densities required for high speed and high gain, as well as for reliability, as most degradations worsen at higher temperatures. For relatively lower power, the junction temperature rise (Trise) is typically modeled as a linear function of the dissipated power (Pdiss), with the slope being a parameter named thermal resistance (Rth), which can be further modeled as a function of ambient temperature (Tamb).
In Mextram, this was the case until the recent implementation of the nonlinear SH model of [1]-[4] in Mextram 505.4 to improve modeling for high-power applications [5] [6]. A notable feature of the Mextram nonlinear SH model implementation is that it uses the same model parameters as its linear SH model, i.e., the low power Rth at reference temperature, rth, and the material thermal conductivity temperature coefficient ath used to describe the Tamb dependence of the linear SH Rth. In the nonlinear SH model, the same ath is used for Pdiss dependence of the effective Rth.
Theoretically, if ath is already physical, one can simply turn on the nonlinear SH switch for improved modeling at higher collector current (IC) and collector-emitter voltage (VCE). The key to this simplicity lies in the specific form of the nonlinear functional dependence of Trise on Tamb and Pdiss derived by solving the nonlinear heat diffusion equation using the Kirchhoff transformation [7].
In practice, however, retuning these two parameters with high power data is beneficial for better overall fitting, particularly over multiple ambient temperatures. Fitting a larger power range at multiple ambient temperatures can still be challenging, due to the presence of multiple materials with different thermal conductivity temperature coefficients in the heat flow path and heat dissipation through multiple paths, e.g., from metallization in addition to substrate.
To provide flexibility in silicon data fitting and extend the model's capability to a wider ambient temperature range and power range, we present an additional semi-empirical model. By deriving the slope of Trise versus Pdiss in the single material single path model problem, we show that it is only a function of the junction temperature Tjunc=Trise+Tamb, which is behind the ability of the model to use only two parameters for both Tamb and Pdiss variation.
We propose a semi-empirical modification of the Trise versus Pdiss slope behavior, similar to the low-temperature CMOS measurement-based model in [8] and [9], which has the same issue of having a Tjunc-dependent Trise vs Pdiss slope, but an easier functional form for modification to provide flexibility in better fitting high power region over a wider range of ambient temperature. Our model construction ensures that the low Pdiss behavior is guaranteed to be the same as existing Mextram SH models.
Figs. 1-3 show the modeling results of IC-VCE at various base-emitter voltages (VBEs) for a SiGe HBT at -25, 25, 75, and 125°C using 1) Mextram's default linear SH model; 2) Mextram 505.4's nonlinear SH model; and 3) the proposed wide power and temperature range nonlinear SH model. The new model shows the best overall fitting.
In the full paper, we will also present results obtained using the nonlinear SH model used in HICUM L2 [10], where the effective Rth modeled as a linear or nonlinear function of Tjunc, and results obtained by implementing the model of [9]. We will make comparisons of the various models and provide best practice recommendations.
[1] W. B. Joyce, SSE, vol. 18, pp. 321-322, 1975.
[2] K. Poulton et al., JSSC, vol. 27, no. 10, pp. 1379-1387, 1992.
[3] B. Yeats, Dig. GaAs IC symposium, pp. 59-62, 1999.
[4] J. C. J. Paasschens et al., BCTM, pp. 96-99, 2004.
[5] G. Niu et al., The Mextram Bipolar Transistor Model Version 505.4.0, 2023.
[6] M. Willemsen, AKB-Workshop, 2023.
[7] G. R. Kirchhoff, Vorlesungen über die Theorie der Wärme, 1894.
[8] K. Triantopoulos et al., TED, vol. 66, no. 8, pp. 3498-3505, 2019.
[9] G. Ghibaudo et al., SSE, vol. 192, article 108265, 2022.
[10] M. Schroter, HICUM/L2 Technical documentation of model version 3.0.0, 2020.

Figure 1