# Development of Highly Sensitive Temperature Microsensors for Localized Measurements
## 2 Materials and Methods
### Numerical Simulations  
In order to study the performance of the several temperature microsensors with different number of windings (5, 8, 12, 16, 20, 24), numerical simulations using the finite elements method (FEM) in COMSOL Multiphysics 5.3 were performed. The model consists of a platinum-based microsensor with a meander-shaped geometry in a silica glass substrate. The numerical simulations started with the study of the self-heating using several excitation currents. This effect, which depends on the current value that pass through the microsensor, introduces drift errors in the measurement and reduces the system accuracy. For this study, a fixed ambient temperature was set to the model and the electric potential \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Windings in the** & **Width (\(\upmu\)m)** & **Length (\(\upmu\)m)** & **Area (\(\upmu\)m\({}^{2}\))** \\
**Meander-Shaped Geometry** & & & \\ \hline
5 & 400 & 190 & 76,000 \\
8 & 400 & 310 & 124,000 \\
12 & 400 & 470 & 470,000 \\
16 & 400 & 630 & 252,000 \\
20 & 400 & 790 & 316,000 \\
24 & 400 & 950 & 380,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical dimensions of each fabricated sensor. Figure 2: Schematic representation of the overall geometry of the temperature microsensors. and the self-heating generated by the sensor, when excited by an electric current, was outputted. To achieve this, the physic models \"Heat Transfer in Solids\" and the \"Electric Currents, Shell\" were used alongside the multiphysics module \"Electromagnetic Heating\" to address the joule heating effect, responsible for the sensors self-heating. The heat transfer in solids is governed by the main Equations (1) and (2):  
\[\rho C_{p}\frac{\partial T}{\partial t}+\rho C_{p}u\cdot\
abla\cdot q=Q \tag{1}\]  
\[q=-k\
abla T \tag{2}\]  
where \(\rho\) (SI unit: kg/m\({}^{3}\)) is the solid density, \(C_{p}\) (SI unit: J/(kg\(\cdot\)K)) is the solid heat capacity at constant pressure, \(k\) (SI unit: W/(m\(\cdot\)K)) is the solid thermal conductivity, \(u\) (SI unit: m/s) represents the velocity field, \(T\) (SI unit: K) is the absolute temperature, \(q\) is the heat flux by conduction (SI unit: W/m\({}^{2}\)) and \(Q\) (SI unit: W/m\({}^{3}\)) is the heat source (or sink). A fixed temperature of interest was attributed to the bottom of the substrate and a resistive thin layer with 200 nm of thickness was assigned to the platinum geometry domain. For the electric currents interface, the main governing Equations (3)-(5) are:  
\[\
abla\cdot J=Q_{j} \tag{3}\]  
\[J=\sigma E+J_{e} \tag{4}\]  
\[E=-\
abla V \tag{5}\]  
where \(\sigma\) is the electrical conductivity (SI unit: S/m), and \(J_{e}\) is an externally generated current density (SI unit: A/m\({}^{2}\)). \(E\) is the electric field (SI unit: V/m) and \(Q\) is the electric charge (SI unit: C). This physics interface was only applied at the platinum domain since the electric conduction at the substrate was neglectable, and for simplification purposes was not considered. The thickness of the shell was defined at 200 nm. To model the resistivity variation of the platinum thin film with the temperature, the electric conductivity is governed by the following Equation (6):  
\[\sigma=\frac{1}{\rho_{0}(1+\alpha(T-T_{0}))} \tag{6}\]  
where \(\rho_{0}\) is the resistivity (SI unit: \(\Omega\cdot\)m) at the reference temperature \(T_{0}\) (SI unit: K), and \(\alpha\) is the temperature coefficient of resistance (SI unit: 1/K), which describes how the resistivity varies with temperature. A resistivity (\(\rho_{0}\)) of 1.2447 \(\times\) 10\({}^{-7}\)\(\Omega\cdot\)m was assigned to the platinum for a reference temperature (\(T_{0}\)) of 20 \({}^{\circ}\)C, as measured by experimental results, and a temperature coefficient of resistance (\(\alpha\)) of 0.00385 was chosen, because of its worldwide adoption and availability in both platinum wire-wound and thin films [26]. An excitation current and a ground condition were defined at the external pads.