1. Uncooled IR Microbolometer Designs

The a-Si uncooled IR sensor was fabricated; Fig. 2 illustrates the (a) top view, (b) cross-view, (c) current flow in the top view, and (d) cell array of 35 ${\mathrm{\mu}}$m pixels. The pixel was designed three-dimensionally and simulated. The thicknesses of the cantilever, the a bsorption layer, the resistance layer, and the passivation layer were 120 nm SiO$_{2}$, 15 nm TiN, 100 nm a-Si, and 110 nm SiO$_{2}$, respectively. The cells were arranged in an 80${\times}$80 array and packaged in a vacuum.

Fig. 2. Uncooled IR microbolometer cell structure (a) Top view, (b) cross-view, (c) current flow in top view, (d) SEM of fabricated 80${\times}$80 cell array with 35 ${\mathrm{\mu}}$m pixel. In this study, the length of the resistance layer and width of the leg were chosen as factors affecting the time constant.

2. Simulation Models

The IR microbolometer measures the current change due to the joule heating by the driving voltage and the heat of the IR from outside. In this study, temperature increase by joule heating was simulated where the time constant was extracted. The time constant indicates how quickly the system can respond to temperature changes ⁽¹²⁾. In the microbolometer, the time to reach 63.2% of the maximum temperature was set as the time constant. The equation is:

$$\tau =C/G,$$

where ${\tau}$ is the time constant, $\textit{C}$ is the thermal heat capacity, and $\textit{G}$ is the thermal conductance. Table 1 presents Specific heat capacity and thermal conductance of materials used in the simulation.

The joule heating is simulated using an Electric Currents module, Heat Transfer in Solids module provided by the COMSOL and the formula is like below ⁽¹³⁾:

$$\rho C_{p}\left(\frac{\delta T}{\delta t}+u_{\textit{trans}}\cdot \nabla T\right)+\nabla \cdot \left(q+q_{r}\right)=- \alpha T\colon \frac{dS}{dt}+Q,$$

where ${\rho}$ is the density of the material, $\textit{C}$$_{p}$ is its heat capacity, $\textit{T}$ is the temperature, $\textit{u}$$_{trans}$ is the velocity vector of translational motion, $\textit{q}$ is the heat flux by conduction, $\textit{q}$$_{r}$ is the heat flux by radiation, $\textit{S}$ is the second Piola-Kirchhoff stress tensor, and $\textit{Q}$ is the heat sources. Furthermore, $\textit{q}$ represents the flow of heat by conduction and is expressed by Fourier’s law as $q=- k\nabla T$, where $\textit{k}$ is thermal conductivity ⁽¹³⁾. $\textit{Q}$ is a term that includes heat generated by other factors and changes in temperature due to conduction, convection, and radiation, which are heat transfer methods. Therefore, $\textit{Q}$ represents the current density, and the current density can be expressed by the following equations ⁽¹³⁾.

$$\begin{equation*} Q=J\cdot E \end{equation*}$$

$$J=\left(\sigma +\mathit{\int }_{0}\varepsilon _{r}\frac{\delta }{\delta t}\right)E+J_{e},$$

where J$_{\mathrm{e}}$ is the current density applied externally and defaults to 0. In the IR microbolometer, the change in the current by the temperature change is detected and so the ${σ}$ value is set as a parameter related to the temperature. The equation is ⁽¹³⁾:

$$\sigma =\frac{1}{\rho _{0}\left(1+\alpha \left(T- T_{0}\right)\right)},$$

where ${\rho}$$_{0}$ is the specific resistance at $\textit{T}$$_{0}$, ${\alpha}$ is the temperature coefficient of resistance (TCR), $\textit{T}$ is the heated temperature, and $\textit{T}$$_{0}$ is the reference temperature, usually 20$^{\circ}$C.

The heat transfer in a solid is ⁽¹³⁾:

$$\begin{equation*} \rho C_{p}\frac{\delta T}{\delta t}=k\nabla \cdot \nabla T+J\cdot E, \end{equation*}$$

Therefore, when the material eigenvalue is determined as a constant with the applied voltage from the outside, a differential equation relating to the temperature is obtained, and the temperature change over time can be calculated.

In addition to determining IR, we calculated the energy of the wavelength of the IR light absorbed by the IR microbolometer. For the calculation, Wien’s law and Stefan-Boltzmann’s law are used. Wien’s law states that the wavelength at which an object emits maximum radiation energy at a certain temperature is inversely proportional to temperature. The equation is ⁽¹⁴⁾:

$$\lambda _{max}=\frac{2898}{T},$$

where ${\lambda}$$_{max}$ is the wavelength representing the maximum radiation intensity. Stefan-Boltzmann’s law states that the sum of the light energy of all wavelengths emitted from a unit surface area of a black body is proportional to the fourth power of the absolute temperature of the black body ⁽¹⁴⁾:

$$P=\varepsilon \sigma eT^{4},$$

where ${\sigma}$ is the Stefan-Boltzmann constant and ${\varepsilon}$ is emissivity. Emissivity refers to the ratio of the radiant intensity of an object with the same temperature to the blackbody surface.

3. External Environment Setting

An important factor for accurate time constant simulation is the external energy loss of the cell. Fig. 3 is a simple schematic diagram of the heat loss that can occur in the cell.

Fig. 3. Heat loss model in IR microbolometer cell. Q$_{\mathrm{Sink}}$ is heat release by heat transferred to the ROIC substrate through the anchor and Q$_{\mathrm{Convection}}$ is heat release by ambient convection.

If external energy loss is not considered in the simulation, the temperature of the cell tends to continuously increase during operation because there is no heat emitted to the outside and the heat energy inside accumulates. Therefore, to extract the time constant, it is necessary to set the supplied thermal energy and the lost thermal energy (Q$_{\mathrm{Exernal}}$). The supplied thermal energy is determined by the sum of the thermal energy by Joule heating (Q$_{\mathrm{Joule heating}}$) and IR (Q$_{\mathrm{IR}}$). The total heat energy (Q$_{\mathrm{Total}}$) applied to the cell can be expressed as:

$$Q_{\text {Total }}=\left(Q_{\text {Joule heating }}+Q_{I R}\right)-Q_{\text {External }}$$

where $\textit{Q}$$_{External}$ can be expressed as the sum of the heat release due to convection to the outside (Q$_{\mathrm{convection}}$) and the heat release to the ROIC substrate through the anchor (Q$_{\mathrm{Sink}}$):

$$Q_{\text {Extemal}}=Q_{\text {Convection}}+Q_{\operatorname{Sink}}$$

However, the IR microbolometer is vacuum-packaged, so convection with the outside does not occur. In this study, the boundary condition is set such that there is no heat flow except where ROIC substrate and anchor are bonded. Accordingly, only Q$_{\mathrm{sink}}$ was considered in this study. For the simulation, Q$_{\mathrm{Sink}}$ and Q$_{\mathrm{IR}}$ are extracted from the fabricated device with values of 6350 W/(m$^{2}$·k) and 976 W/m$^{2}$. Q$_{\mathrm{IR}}$ and Q$_{\mathrm{Sink}}$ were set based on a 35 ${\mathrm{\mu}}$m pixel, and the values were normalized to the area even if the cell size is changed by downscaling, the conditions of the simulation are the same.

4. Temperature Coefficient of Resistance (TCR) Setting

Another important factor for accurate time constant analysis is the TCR of a-Si. TCR of a-Si is known to be temperature-dependent but in many cases, the TCR is assumed to be a constant. To improve simulation accuracy, the temperature-dependent TCR of a-Si is used in this study. Fig. 4 illustrates the TCR based on the temperature; the black curve represents the experimental data presented in a reference paper ⁽¹⁴⁾ while the red curve is fitted to model the TCR values of a-Si. In the reference paper, TCR was extracted based on temperature using the manufactured a-Si device. In this study, TCR was expressed as a function of temperature and applied to the simulation.

Fig. 5 illustrates the simulation results of the cell current according to the time during the operation considering the factors described above. The applied voltage is 1.8 V in the simulation. Comparing the simulation with the measurement values of the fabricated 35 ${\mathrm{\mu}}$m pixel devices, the two current values are nearly equivalent. Based on these results, the applied voltages to the standard designed cells of 25 and 17 ${\mathrm{\mu}}$m were set to the current between 1 and 2 ${\mathrm{\mu}}$A and 1.3 and 1 V. The standard dimensions of the resistance layer and leg for each pixel size are depicted in Table 2.

Fig. 4. TCR of a-Si used in the simulation. The black curve represents experiment data ⁽¹⁵⁾ and the red curve is fitted to model the TCR.

Fig. 5. Simulation results of the cell current based on the time during the operation with the measurements in the fabricated 35~${\mathrm{\mu}}$m cell.