^{1} Thus, an accurate large-signal model for HBT devices is of great importance in designing such circuits, especially when the transistor is operated in nonlinear regions where self-heating effects become significant.

Although the temperature sensitivity of transistor parameters is significant for all types of power transistors, it is particularly important for the HBT, with its relatively poor thermal conductivity.

It is well known that the saturation currents and ideality factors of model diodes change with ambient temperature. This change has been fitted to exponential and polynomial functions in HBT models.^{2} However, if it is assumed that the saturation currents and ideality factors remain fixed at their ambient temperature values, their associated temperature dependence can be attributed to the increase of the base-emitter voltage by an incremental thermal voltage V_{TH}.^{3} This is illustrated as follows:

where

V_{be0} = base-emitter voltage at ambient temperature T_{0}

P_{dsp} = dissipated power in the device

R_{th} = thermal resistance

#### Determination of Thermal Resistance

The proposed method of determining thermal resistance uses forward Gummel measurements for different ambient temperatures and for different collector-emitter bias voltages V_{ce}. This method was applied to an on-wafer 2 x 25 mm^{2} GaInP/GaAs HBT in a common-emitter configuration. The DC measurements were accomplished using a Cascade probe station, monitored by a program developed using HP-VEE software. A thermal chuck was used to set the ambient temperature of the device. MATLAB software was used for programming the extraction of the thermal resistance.

As proven by Zhang et al.,^{4} for a constant emitter current I_{E}, the base-emitter voltage V_{be} varies linearly with the junction temperature T_{j}. Thus, around an arbitrary temperature T_{1}(T_{1}≥T_{0}), the voltage V_{be} at temperature T_{j} can be written as

where

V_{be1} = base-emitter voltage at temperature T_{1}

Knowing that T_{j} = T_{0} + R_{th}P_{dsp}, Equation 4 becomes

Two sets of measurements are necessary to determine the thermal resistance R_{th}. The first set of measurements contains the Gummel data at a fixed collector-emitter voltage of 1.5 V and variable ambient temperature (see *Figure 1*). The second set of measurements contains the Gummel data for a fixed ambient temperature (25°C) and a variable collector-emitter voltage. The extraction of the thermal resistance is illustrated in the following two steps:

**Step 1**

Considering the parameter P_{1} as the slope of the linear variation of the voltage V_{be} versus the temperature T_{1} while the dissipated power P_{dsp} is maintained constant, as shown in *Figure 2*, one can write

The dissipated power was calculated from: P_{dsp} = I_{c}V_{ce} with I_{c} ª I_{E} since I_{c} > 100I_{b} in the considered bias cases. The values of I_{c} and V_{ce} were determined at a 25°C ambient temperature in each case, for the considered emitter current I_{E}. In this first step, a set of values of the parameter

for different dissipated power is obtained.

**Step 2 **

Considering the parameter P_{2} as the slope of the linear variation of the base-emitter voltage V_{be} versus the dissipated power P_{dsp} (corresponding to a constant emitter current I_{E}) while maintaining the temperature T_{1} constant, one can write

The dissipated power was calculated from P_{dsp} = I_{c}V_{ce} with I_{c} ª I_{E}.

The values of I_{c} were determined at a 25°C ambient temperature in each case of the considered constant emitter current I_{E} (see *Figure 3*). The collector-emitter voltage was fixed at 1.5 V. The values of the emitter current I_{E} were the same as those considered in the first step. In this second step, a set of values of parameter

for different dissipated powers was obtained. The used values of the dissipated powers were the same as those determined in the first step.

Finally, using the values of parameters P_{2} and P_{1}, the ratio P_{2}/P_{1} was calculated, which allows for determination of the thermal resistance at each considered dissipated power P_{dsp} and emitter current I_{E}

Experimental validation showed that the determined thermal resistance varies linearly versus the dissipated power. As shown in *Figure 4*, this variation is linear and can be calculated using the following derived formula

#### Inclusion of Thermal Effect in a HBT DC Model

The determined R_{th} was used to calculate dynamically the self-heating in a developed HBT DC model. A set of parameters for this model was determined, using methods reported previously.^{5,6} The proposed HBT DC model was implemented in the commercial simulator ADS as a symbolically defined device (SDD).

The implemented electro-thermal model has predicted accurately the DC characteristics of a 2 x 25 mm^{2} InGaP/GaAs HBT device, as shown in *Figure 5*.

#### Conclusion

A practical and detailed method for accurate determination of thermal resistance R_{th} and its power dependence was developed. This method used forward Gummel data at different ambient temperatures and at different collector-emitter bias voltages. The linearly power dependent thermal resistance was successfully used to calculate the junction temperature variation in a DC model of a 2 x 25 mm^{2} GaInP/GaAs HBT device. n

#### Acknowledgment

This research work was conducted as part of the author’s PhD studies at the PolyGrames Research Center, Ecole Polytechnique de Montreal, Quebec, Canada.

#### References

- P.M. Asbeck, M.C.F. Chang, J.A. Higgins, N.H. Sheng, G.J. Sullivan and K.C. Wang, “GaAlAs/GaAs Heterojunction Bipolar Transistors: Issues and Prospects for Application,” IEEE Transactions on Electron Devices, Vol. 36, No. 10, October 1989,

pp. 2032–2042. - C.T. Dikmen, N.S. Dogan and M.A. Osman, “DC Modeling and Characterization of AlGaAs/GaAs Heterojunction Bipolar Transistors for High Temperature Applications,” IEEE Journal of Solid-State Circuits, Vol. 29, No. 2, February 1994,

pp. 108–116. - Q.M. Zhang, H. Hu, J. Sitch, R.K. Surridge and J.M. Xu, “A New Large-signal HBT Model,” IEEE Transactions on Microwave Theory and Techniques, Vol. 44, No. 11, November 1996, pp. 2001–2009.
- D.E. Dawson, A.K. Gupta and M.L. Salib, “CW Measurement of HBT Thermal Resistance,” IEEE Transactions on Electron Devices, Vol. 39, No. 10, October 1992,

pp. 2235–2239. - S. Bousnina, P. Mandeville, A.B. Kouki, R. Surridge and F.M. Ghannouchi, “Direct Parameter-extraction Method for a HBT Small-signal Model,” IEEE Transactions on Microwave Theory and Techniques, Vol. 50, No. 2, February 2002, pp. 529–536.
- S. Bousnina, C. Falt, P. Mandeville, A.B. Kouki and F.M. Ghannouchi, “An Accurate On-wafer De-embedding Technique with Application to HBT Devices Characterization,” IEEE Transactions on Microwave Theory and Techniques, Vol. 50, No. 2, February 2002, pp. 420–424.

**Sami Bousnina** received his PhD degree in electrical engineering from the Ecole Polytechnique de Montreal, Quebec, Canada, in 2004. In 1999, he was an intern with Nortel Networks, Ottawa, Ontario, Canada, working on characterization and modeling of HBT devices. From 2004 to 2005, he was with Advanced Power Technology RF as a senior development engineer. He is currently a senior RF design engineer with M/A-COM, a division of Tyco Electronics, responsible for silicon bipolar device modeling and RF pulsed amplifier design.