large power device.
To prevent the thermal instability of multi-finger transistors, ballasting resistors are often used. The voltage drop across these resistors compensates the build-in voltage change due to temperature rise caused by self-heating and as a result the thermal sta-bility is improved. Traditionally, the fingers and the ballasting resistors connected to the fingers are usually identical to one another. However, because of the nonuniform heat dissipation, it has been realized that the uniform layout commonly used for the fin-gers of power transistor is not ideal for thermal stability. A non-optimized design can easily over correct the problem and even make the problem worse. If we over design the ballasting resistors, the performance of the device will suffer. For this reason, an optimal design of ballasting resistors is very important. We need to know an minimal value of ballasting resistors to just make the device stable. In this work, we tried to find the optimum values of the ballasting resistors for each finger. And the optimum design procedure for thermally stable multi-finger transistor was developed.
1.3 Overview and Outline
In Chapter 2, we first discuss the thermal resistance which is the origin of the ther-mal feedback phenomenon of semiconductor devices. Although it is usually treated as a fitting parameter and without any explanation, it has more physical significance than just a fitting parameter and is derivable from the fundamental heat flow equation. The analysis of this work is based on the assumption that the thermal resistance matrix is already known. This chapter will talk about the methods to obtain the thermal resis-tance and discuss its properties. Both of the theoretical and experimental methods to determine the thermal resistance will be investigated and compared. Based on the re-sults, some useful remarks about the chip layout design to reduce the thermal resistance will be given. Additionally, when the temperature dependent thermal conductivity is taken into account, by the help of Kirchoff transform, the thermal resistance obtained
1.3 Overview and Outline
by solving the temperature independent kth heat flow equation can be used to rewrite the temperature equation of a device from a linear function of the dissipated power to a nonlinear one. The theoretical background of the extraction methods will be explained by the temperature dependent thermal conductivity effect.
In Chapter 3, the temperature dependence of the current-voltage equation is taken into account. This interaction between current and temperature is called self-heating because the device temperature usually becomes higher than that without this effect.
The self-heating effect will make the current-voltage equation nonlinear and couple the current-voltage behavior of each finger together. Because of the thermal coupling effect, the finger temperature and the finger current will affect each other. The coupled current-voltage expressions of each finger are functions of other finger currents. The coupled current-voltage equations in simple model and accurate model was defined and derived in this chapter. The method to solve the coupled current and temperature simultaneously was also discussed. The MATLABr[6] program code used to solve the coupled current-voltage equations was listed in Appendix A.
In Chapter 4, we solved the basic coupled current-voltage equations in simple model.
We found that there is an ideal distribution for the ballasting resistance. Significant im-provement in thermal stability can be obtained when the ideal distributions are used.
Simple analytical formulas for the ideal distributions of the emitter ballasting resistance to achieve the highest stable operation current are derived by using the concept of can-cellation of positive and negative feedback. With the ideal distribution, we have also found an optimum emitter ballasting resistance for absolutely thermal stable operation condition that the device never becomes unstable. Base on the above results, a design procedure for multi-finger transistors is developed [7].
In Chapter 5, we extended our model to the accurate model and take the temperature dependence of thermal conductivity into account. A more physical model will give us a more practical result. By this manipulation, although the behavior of the current-voltage curves are quit different from the simple model, the basic concept of the ideal ballasting
1.3 Overview and Outline
resistance distribution and the absolutely stable condition are still similar and only need small modification. We could use the same derivation procedure to derive the ballasting resistance equations for the accurate model. Two practical design procedures based on uniform current and uniform temperature consideration were obtained and described from the accurate model [8]. The results with the temperature dependent and constant thermal conductivity were also compared.
Chapter 2
The Thermal Resistance
T
he thermal resistance is the central parameter to describe the thermal behaviors of semiconductor devices. This important parameter, which is usually treated as a fitting parameter and without any explanation, is the origin of the thermal feedback phenomenon of devices. It has more physical significance than just a fitting parameter and is derivable from the fundamental heat flow equation. The analysis of this work is based on the assumption that the thermal resistance matrix Rth is already known.Therefore, this chapter will discuss the thermal resistance further in advance, including the methods to obtain it and its properties. There are two main categories of methods to determine the thermal resistance Rth, theoretically and experimentally. Both of these two methods will be investigated and compared in this work. The theoretical result will show that Rth is a function of device geometry and inverse proportional to the thermal conductivity kth of substrate wafers. Based on this result, some useful remarks about the chip layout design to reduce Rth will be given.
Additionally, when the temperature dependent thermal conductivity is taken into account, by the help of Kirchoff transform, the thermal resistance obtained by solving the temperature independent kthheat flow equation still served as an essential parameter.
The only change needed in this case is that one has to rewrite the temperature equation of a device from a linear function of the dissipated power to a nonlinear one. Of course,
2.1 Thermal Conductivity of GaAs and InGaP
this change will affect device thermal behavior substantially but it will not increase too much mathematical efforts. The following discussion will include this temperature dependent thermal conductivity effect to understand the theoretical background of the extraction methods.
2.1 Thermal Conductivity of GaAs and InGaP
The thermal conductivity is a basic material parameter. It is the key factor of ther-mal behaviors of all devices. Although the temperature dependent property of GaAs has been revealed for decades, a large discrepancy of data between literatures is still a problem. Anholt has collected several expressions and values of the GaAs thermal con-ductivity from different references (Table 4.1 in [9]). The deviations of parameters used by different authors are remarkably. The physical preferable temperature dependence expression could be written as [10], [11]
kth(T ) = kth0 µ T
300
¶−b
. (2.1)
This expression has the mathematical advantages of integration and differentiation with respect to temperature. Table 2.1 summarizes the values used in this work and the values of silicon [12] and gold for reference. The experiment data which could be used to determine the parameters in this expression can be found in [13]-[16]. Because that HBT devices are always fabricated on semi-insulating substrate wafers, Blanc’s
kth0(W/cm-◦C) b
GaAs 0.45 1.25
InGaP 0.05 0
Si 1.56 1.31
Au 3.17
-Table 2.1. The thermal conductivity of GaAs, InGaP, Si and gold.