The Multiphysics tool is used to design the multilayer Al/Graphene/SiO2/Si temperature sensor geometry structure shown in Fig. 1.a. The graphene layer acted as a piezo resistor. The top and bottom layers are Aluminium and monocrystalline silicon, respectively. The SiO2 acts as a composite layer between graphene and aluminium. The designed piezoresistive temperature sensor has four piezoresistive elements and is connected according to the Wheatstone bridge and the meshed structure shown in Fig. 1. b.

It is reported that the thermal expansion coefficient for the single layer graphene is 3.2*10− 6 K-1 when temperature variations are used to obtain the desired strain field [13]. At electron densities of 2*1011 cm2, electron mobilities of 200,000 cm2 V1 s1 [14]. The thermal conductivity, density, and young modulus are 5000 Wm-1K-1, 0.77 mgm-3, and 1100 GPa [15]. When the substrate is stressed, the material is not easily damaged so that a stable conductive network can be formed. We define the matrices W, X, Y, Z, J and K as

W=\(\left|\begin{array}{cccc}\frac{1}{{E}_{Si}{t}_{Si}}& \frac{1}{{E}_{Gr}{t}_{Gr}}& 0& 0\\ 0& \frac{1}{{E}_{Si02}{t}_{Si02}}& \frac{1}{{E}_{Gr}{t}_{Gr}}& 0\\ 0& 1& \frac{1}{{E}_{Gr}{t}_{Gr}}& \frac{1}{{E}_{Al}{t}_{Al}}\\ 1& 1& 1& 1\end{array}\right|\) [1]

X=\(\left|\begin{array}{c}{t}_{Si}+{t}_{SiO2}\\ {t}_{SiO2}+{t}_{Gr}\\ {t}_{Gr}+{t}_{Al}\\ 0\end{array}\right|\) [2]

Y=\(\left|\begin{array}{c}{\alpha }_{Si}-{\alpha }_{SiO2}\\ {\alpha }_{SiO2}-{\alpha }_{Gr}\\ {\alpha }_{Gr}-{\alpha }_{Al}\\ 0\end{array}\right|\) [3]

Z\(={\left({\sum }_{L=1}^{4}Ei{I}_{i}\right)}^{-1}\) \(\left|\frac{{t}_{Si}}{2}+{t}_{Si}+\frac{{t}_{Sio2}}{2} {t}_{Si}+{t}_{SiO2}+\frac{{t}_{Gr} }{2} { t}_{Si}+{t}_{SiO2}+{t}_{Gr}+\frac{{t}_{Al}}{2}\right|\) [4]

J=\(2\frac{\text{Z}{\text{W}}^{-1}\text{Y}}{\text{Z}{\text{W}}^{-1} \text{X}-2}\) [5]

K=\(2\frac{\text{Z}{\text{W}}^{-1}\text{Y}{\text{W}}^{-1}\text{X}}{\text{Z}{\text{W}}^{-1} \text{X}-2}-{\text{W}}^{-1}\text{Y}\) [6]

Where ESi, ESiO2, EGr, and EAl are the young moduli of silicon, graphene, and aluminium. The thicknesses of Si, SiO2, Graphene, and Aluminium layers are represented by tSi, tSiO2, tGr, and tAl, respectively, and the thermal expansion coefficients of silicon, SiO2, Graphene, and Aluminium are represented by Si SiO2 Gr and Al.

σ = \(\frac{\text{K}\varDelta \text{T}}{{t}_{Gr}}+\)JE∆T\(\left(\frac{{t}_{Gr}}{2}-d\right)\frac{{t}_{Gr}}{2}\) [7]

Just the young modulus, thermal expansion coefficient, and thickness affect the values of J and K.

∆R1=∆R3= -∆R2=-∆R4= \(\frac{\text{R}{\Pi }44 {\sigma }}{2}\)[8]

The output voltage (Vo) can be given as below:

Vo= \(\frac{{\text{V}}_{in}{{\Pi }}_{44} {\sigma }}{2}\) [9]

σ is the thermal-stress, variations of piezo resistances are called as ∆R1, ∆R2, ∆R3 and ∆R4, π44 is called as temperature of the piezo resistance coefficient. The sensitivity of a temperature sensor is defned as the ratio of change in the output voltage to the change in the applied temperature with the input voltage. Mathematically it is defned by

Sensitivity= \(\frac{\varDelta \text{V}}{\varDelta T}\). \(\frac{1}{{V}_{in}}\) [10] The piezo resistance relative change rate of the temperature sensor denoted as

\(d\left(\frac{\varDelta R}{R}\right)/dT=d\left(\frac{{V}_{o}}{{V}_{in}}\right)/dT\) [11]

Where ∆R is the change of piezo resistance due to deformation. \(dT\) is the rate of change in temperature. From Eq. 1 to Eq. 11 are used to for simulation of piezo resistance temperature sensor