Degradation of Electrical Parameters of Power Semiconductor Devices – Process Influences and Modeling
4.3 Reliable device temperature extrapolation
One of the main benefits of the polyheater is that very high temperatures can be achieved by supplying large electric power to the heater. However, the methods to determine the device temperature presented in the preceding Section can only be used up to the highest temperature of an external heating system like the thermal chuck or a dedicated furnace. For higher temperatures an extrapolation scheme is needed. One could, in principle, extrapolate the relationship to higher values with the help of a technology computer aided design (TCAD) simulation. However, for this approach calibrated material parameters for the device are needed. Those parameters need to be measured on a test system which can handle such high temperatures. Additionally, at these high temperatures already the biasing for the dependent drain current measurement can lead to degradation of the device. To overcome these limitations an extrapolation scheme was developed which has the potential to work up to arbitrary temperatures. Only the breakdown field of the field oxide and electromigration in the poly wires should create a limit before the materials which build up the device may melt [PobegenTDMR13]. This extrapolation method uses the change of the thermal resistance of the substrate to calculate the device temperature directly from the power dissipated in the heater.
4.3.1 Thermal resistance measurement
At the heart of the extrapolation method lies the idea that the change of the thermal resistances between the device and the heat sink with temperature needs to be taken into account. The thermal resistance is defined as
For Joule heating, the heat flow equals the dissipated electric power and leads to a temperature rise . Consequently, an apparent thermal resistance can be calculated from the rise of the temperature with . As a particular example, in Fig. 4.5 the device temperature rises by about 30 °C with 1 W of power supplied to the polyheater. This means that the apparent thermal resistance of the substrate below the device , which includes the threedimensional (3D) heat spread in a phenomenological fashion, is about 30 °C/W at 30 °C chuck temperature. If this measurement is now repeated at several chuck temperatures the experimental temperature dependence of the substrate thermal resistance is obtained. In Fig. 4.6 the values for several technologies are drawn.
As can be seen, for the value of the type of transistor (n or pMOSFET), its area and the type and thickness of the gate oxide are irrelevant. Also devicetodevice variations are very small. The only important parameters are the type and thickness of the substrate material [PobegenTDMR13]. For all investigated technologies the dependence of the thermal resistance on the temperature can be approximated by a linear function in the investigated temperature range, as can be seen from the dashed lines in Fig. 4.6.
4.3.2 Analytical extrapolation method
Using the definition of the thermal resistance
and an approximately linear dependence of on the temperature
one obtains the differential equation
With the requirement that the device temperature equals the chuck temperature when there is no power supplied to the heater ( ), the equation has the solution
This means that the rise of the device temperature is an exponential function of the heater power [PobegenTDMR13; Dar+12]. For the standard formulation of the thermal resistance with the three constants , and
where is the temperature dependence of and is the reference thermal resistance at temperature , the solution is
or, if
With this equation it is possible to calculate the temperature of the DUT directly from the power dissipated in the polyheater. This method has several advantages:

• The extrapolation depends on a linear fit of which is a steady state measurement and can be performed with great accuracy.

• The approach is independent of devicetodevice variations because the drain current is only needed once to determine . Repeated use of the same parameters on different devices are independent of the characteristics of the actual device. Only the structure and dimensions of the polyheater must be the same.

• The electrical resistance of the polyheater can vary within a wafer or a technology because of process variations for the poly deposition. The proposed method is independent of such heatertoheater variations since the biasing of the polyheater is adjusted to ensure a constant power dissipation which means a constant heat generation.

• The measurement of needs to be done only once per technology and not once per DUT as for the conventional method described in Section 4.2.1.

• The method is very accurate and precise in the temperature range where it can be compared to measurement data, i.e. in the range of the thermal chuck, cf. Fig. 4.10.

• The extrapolation depends only on the assumption that the temperature dependence of the thermal resistance will not change its behavior in the extrapolated region. This is a fairly save assumption considering that the temperature dependence between 0 K and the melting point of Si is well captured by [Let+87]. Using exactly this dependence, however, leads to a solution of the differential equation (4.4) as
which is rather annoying to handle. But the temperature dependence of can be approximated reasonably well by a linear function in the range of the polyheater use which gives also very accurate results and highlights the essential aspect that the device temperature follows roughly an exponential function on the power supply.

• The approach is independent of the actual material between the device and the heat sink. It can be in principle also be used for other technologies like silicononinsulator, SiC based MOSFETs or gallium nitride (GaN) based transistors.
In other words, the expression (4.7) approximates the complex thermal problem of the device and the polyheater by a simple onedimensional (1D) thermal model as sketched in Fig. 4.7.
The main assumption of this model is that the heat is mainly flowing from the polyheater to the backside of the wafer, which neglects heat transport through the top surface. This is justifiable since heat may leave the top surface only through radiation or convection. Radiation may be estimated by the Stefan–Boltzmann law to be negligibly small ( if ). Convection can be estimated from Newton’s law of cooling with a heat transfer coefficient of air at atmospheric pressure of about 20 Wm^{−2}K^{−1} to 30 Wm^{−2}K^{−1} to be on the order of 10 mW for the small area of a semiconductor test structure [PobegenTDMR13].
The method uses the parameters of , which represent the effective thermal resistance of the substrate at the chuck temperature. However, during polyheater use the substrate experiences a rather large temperature gradient since the top of the substrate is at the device temperature and the bottom at the chuck temperature. The increase of the device temperature with heater power during polyheater use can be expressed by . So this value should be in principle different from the aforementioned value because of the temperature gradient. Still, as shown in Fig. 4.8, the two values coincide.
In the following Section it will be shown, utilizing a 3D electrothermal simulation, that the most probable reason for this is the occurrence of a bottleneck effect.