Analytical model for the electro-thermal feedback effect in a microbolometer infrared focal plane array

Abstract. An analytical model for the electro-thermal feedback effect in a microbolometer infrared focal plane array is presented. The presented model is the integrated optical-electro-thermal model, in which the electro-thermal feedback effect incorporated with the response of incident IR can be described. In addition, since the model is based on physics, the model parameters also have their own physical meaning. This analytical model can be easily utilized to describe the temperature increase caused by the applied heat sources and has a unique feature describing capability of optical-electro-thermal analysis in a quasi-steady-state, which can hardly be performed with thermal analysis tools based on the finite element method. The model shows that the temperature of the microbolometer in this study can be increased 7.1% to 18.6% more by the electro-thermal feedback effect.


Introduction
Uncooled microbolometer infrared focal plane array (IRFPA) technology has extended its application to the small-size, low-weight, low-power, and low-cost IR systems. [1][2][3] As the density of IRFPA has rapidly increased, several technical challenges, such as process technology, optical fill-factor, thermal isolation, and the thermal drift of their characteristics, have emerged. [4][5][6][7][8][9][10][11] The accurate performance prediction of the high-density microbolometer IRFPA is also one of the emerging challenges. [8][9][10][11][12][13][14][15][16] In order to make a reasonable performance prediction of a highly optimized and high-density microbolometer IRFPA, the correct understanding of the electro-thermal phenomena that occur in a microbolometer IRFPA is essentially required. Since the electro-thermal effect is related to the temperature change and the electronic state change of a sensing material, this effect is inherent in a thermal detector like the microbolometer. Hence, the correct understanding of the electro-thermal effect leads not only to a more accurate signal prediction in the development, but also to the proper use of an IRFPA in the application period. [13][14][15][16][17][18] In this article, two analytical models for the electro-thermal feedback effect in the microbolometer IRFPA are presented. The electro-thermal feedback effect starts when the bias is applied to read the resistance change of the microbolometer in IRFPA. Once the bias is applied, the bias heat, which is inversely proportional to the resistance of the microbolometer for a constant voltage bias, is produced in a sensing material. This bias heat leads to a rapid temperature increase in the microbolometer. Since the negative temperature coefficient of resistance (NTCR) of IR sensing materials such as VOx and a-Si is used for most microbolometer technology, the bias heat makes the resistance decrease further. Once again, the further the resistance of the microbolometer decreases, the more the bias heat is produced. Therefore, this electro-thermal process is a positive feedback. The electro-thermal feedback increases the apparent signal of the microbolometer IRFPA. Thus, the measured signal voltage may be larger than that predicted without considering the electro-thermal feedback. In order to reduce this positive feedback effect, the biasing time should be decreased. However, decreasing the bias time, which determines the integration time, is not desirable since it increases the noise bandwidth, and hence the noise equivalent temperature differenc. [15][16][17] As a result, it is of high importance to understand the electro-thermal feedback effect as correctly as possible.
2 Electro-Thermal Feedback Models Since we are interested in the electro-thermal feedback, our primary goal is to find a proper solution to the heat balance equation of a microbolometer in the case, where a constant IR radiation and a voltage bias are applied simultaneously to a microbolometer. The heat balance equation of a microbolometer is given by where H and K are the heat capacity and the thermal conductance, respectively. [13][14][15][16][17][18][19] ΔT ¼ T d − T a . Also, T d , T a , and P are the detector temperature, the constant ambient temperature, and the heat source, respectively. If P is constant in time, the solution to Eq. (1) is given by the following equation: *Address all correspondence to: Seung-man Park, E-mail: smanpark@hoseo.edu where τ th ¼ H∕K is the thermal time constant. The last expression is valid only for t ≪ τ th . The electro-thermal feedback effect is not included in Eq. (2) and this equation has been used to describe the temperature change with applied bias with a slight modification of the bias heat term to be constant as the initial power. We call Eq. (2) the conventional linear (CL) model. As the bias is applied, the P in Eq. (2) is not constant since the bias heat is time dependent. During the constant voltage bias, usually with a pulse form, P can be expressed as follows, in which the bias heat is decomposed into time dependent and independent terms.
where α, ε, and ϕ e are the temperature coefficient of resistance of a sensing material, the IR absorption coefficient, and the incident IR power, respectively. The relationship of R d ¼ R d0 ð1 þ αΔTÞ is also used. Also, the last approximation in Eq. (3) is valid for αΔT ≪ 1. By putting Eq. (3) into Eq. (2), we can get the following equation: From Eq. (4), one can see the feedback characteristic of bias heat. If the incident IR radiation can be negligible, then Eq. (4) becomes the exact same form of a feedback amplifier gain expression 20 with a loop gain of α½ðV 2 Bias Þ∕ðHR do Þt. The only difference is that the loop gain is negative because of the NTCR of the sensing material, so the electro-thermal feedback modeled by Eq. (4) is a positive feedback, as previously mentioned, and is as intuitive as that in the circuit theory. We call Eq. (4), the feedback amplifier gain (FAG) model, for our discussion.
Another approach is possible. With rearrangement after putting Eq. (3) into Eq. (1), one can get an easily solvable equation as follows: where, K eff and P 0 are the effective thermal conductance (ETC) and the time independent heat source, respectively. Whereas the only time independent source term in Eq. (5) with constant incident IR radiation power is treated as a heat source, the time-dependent source term is treated as the term contributing to the ETC. As expected, the solution to Eq. (5) can be found as follows: where τ eff is the effective time constant defined by τ eff ¼ H∕K eff . Although Eq. (8) has the exact same form as Eq. (2), the increasing shapes in temperature are quite different. This difference stems from the fact that K eff is less than K, even can be a negative value, then so do τ eff . The physical meaning of the K eff can be interpreted as follows.
If K eff < 0, then the heat flows into the microbolometer from the ambient. Accordingly, the heat piled up on the microbolometer will rapidly increase, so this piling up of heat will lead to the rapid increase of its temperature. So, ΔT exponentially increased as expected in Eq. (8). If K eff ¼ 0, then the heat flowed out to the ambient is precisely balanced with the heat produced by time dependent bias. Accordingly, the heat piled up on the microbolometer will be increased by an amount of time independent heat source P 0 , so ΔT is linearly dependent on time with a constant heat source P 0 , as in Eq. (5). We will call Eq. (8), the ETC model. Two models were derived from the heat balance equation of a microbolometer. Here, one question arises: Which one is better? In the derivation of Eq. (4), three assumptions were made. The first was t ≪ τ th , the second was αΔT ≪ 1, and the third P was constant in Eq. (2). The first two are very likely. The third is also plausible, in that the mitigation of this assumption was made by quietly inserting the time dependent term of Eq. (3) into Eq. (2). On the other hand, in the derivation of Eq. (8), only the second one was made. So, it can be said that the ETC model is more physical with less assumptions in it. Although Eq. (4) is more intuitive, Eq. (8) is more physical.

Applications of the Models and Discussion
In order to demonstrate the utilization of two analytical models for describing the electro-thermal feedback effect, the FAG model of Eq. (4), the ETC model of Eq. (8), and the CL model of Eq. (2) were applied to a typical microbolometer IRFPA. The characteristics of the microbolometer IRFPA used for this study are summarized in Table 1. [16][17][18][19] The temperatures of the microbolometer calculated by the CL, FAG, and ETC models, respectively, are shown in Fig. 1. It is presumed that the microbolometer is illuminated by an incident IR radiation from a 35°C extended black body source. The inset of Fig. 1 shows the temperature changes of the microbolometer over the first six frame times. Over one frame time of T f , only for the first 100 μs, the bias is applied to the microbolometer, so the temperature rapidly increases for this biasing time. After the biasing time, the microbolometer is cooled down naturally as the bias heat is dissipated to the substrate mainly by conduction. After three to five frame times, the maximum temperature is reached at the end of the biasing time, while the saturated cooling temperature also reached at the end of cooling time. We call this state the quasi-steady-state of a microbolometer. Figure 1 shows the solution to the heat balance equation of the microbolometer with an electro-thermal feedback effect over several frames up to the quasi-steady-state condition. This capability to describe the quasi-steady-state condition is the most unique feature of our model, distinguishing itself from other thermal analysis tools. The temperatures at the end of the biasing time in the quasi-steady-state calculated with the FAG and ETC models are larger than that with CL, in which the electro-thermal feedback effect was not considered. Whereas the temperature change is 8.90 K calculated with CL, those are 10.55 and 9.53 K with the FAG and ETC models, respectively. Thus, it can be seen that the temperatures from the FAG and ETC models are 18.6% and 7.1% higher than that from CL. These results of the electro-thermal feedback effect may be too large to overlook for making more precise tradeoffs among their performance parameters of a high-density microbolometer IRFPA in both the development and application periods.
The modeled resistances of the microbolometer are shown in Fig. 2. The inset of Fig. 2 shows the resistance changes of the microbolometer over the first six frame times like in Fig. 1. In the quasi-steady-state, the resistance changes calculated with the FAG and ETC models are larger than that with CL. While the resistance change is 30.4 kΩ calculated with CL, those are 42.4 and 35.5 kΩ with the FAG and ETC models, respectively. Thus, it can be seen that the resistance changes by considering the electro-thermal feedback effect are 17.3% and 6.7% larger than that from CL. The clear results of the electro-thermal feedback models are shown in Fig. 1 for the temperature and in Fig. 2 for the resistance of the microbolometer. The IR response signals for each temperature of an extended blackbody source can be predicted with the model. [16][17][18][19] The IR response signals can be predicted from the resistance changes calculated with each model. Figure 3 shows the predicted IR response signals of the previous microbolometer IRFPA of Table 1. Here, it is assumed that the microbolometer IRFPA has a columnwise readout with a capacitive transimpedance amplifier as a column amplifier, as indicated in Table 1. 19,21 The IR response signals from the FAG and ETC models, in which the incident IR radiation incorporated with the electro-thermal feedback effect are considered, are larger than that from CL, in which only the IR radiation is considered. This means that the IR response signals are affected by the electro-thermal feedback effect. Since most IR imaging systems utilizing modern IRFPA technologies to get a high quality image are operated after proper nonuniformity corrections (NUC), the relative response signals are more important. The relative temperature responses extracted from the signals of 20°C and 35°C are 19.5, 18.1 mV∕K with the FAG, ETC models, respectively, while 17.9 mV∕K with CL. Compared with the result without an electro-thermal feedback effect, the relative responses with the FAG and ETC models are nearly the same as that with CL. This means that most of the electrothermal feedback effect can be eliminated by NUC. However, the remaining parts still can affect the nonuniformity of an image even after NUC.

Conclusion
The presented analytical model is the integrated opticalelectro-thermal model, with which the electro-thermal feedback effect incorporated with the responsivity of the incident IR can be described. In addition, since the model is based on physics, the model parameters also have their own physical meaning. This analytical model can easily be utilized to describe the temperature and resistance change caused by the applied heat sources and has a unique feature, where it can describe the capability of optical-electro-thermal analysis in the quasi-steady-state, which can hardly be performed with finite element method based thermal analysis tools. The model shows that the temperature of the microbolometer in this study can be increased 7.1% to 18.6% more by the electro-thermal feedback effect.