Silicon Nitride Membrane

Description

Figure 1: silicon nitride membrane temperature profile

A silicon nitride membrane (SiN membrane) ^[1] can be a part of a gas sensor, but also a part of an infra-red sensor, a michrothruster, an optimical filter etc. This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors ^[2] and infrared sources ^[3]. See xx--CrossReference--dft--fig:tempprof--xx, the temperature profile for the SiN membrane.

The governing heat transfer equation in the membrane is:

\nabla \cdot (\kappa \nabla T)+Q - \rho c_p \cdot \frac{\partial T}{\partial t}=0

where $\kappa$ is the thermal conductivity in $W m^{-1} K^{-1}$ , cp is the specific heat capacity in $J kg^{-1} K^{-1}$ , $\rho$ is the mass density in $kg m^{-3}$ and $T$ is the temperature distribution. We assume a homogeneous heat generation rate over a lumped resistor:

Q = \frac{u^2(t)}{R(T)}

with $Q$ the heat generation rate per unit volume in $W m^{-3}$ . We use the initial condition $T_0 = 273K$ , and the Dirichlet boundary condition $T = 273 K$ at the bottom of the computational domain.

The convection boundary condition at the top of the membrane is

q=h(T-T_{air}),

where $h$ is the heat transfer coefficient between the membrane and the ambient air in $W m^{-2} K^{-1}$ .

Discretization

Under the above convection boundary condition and assuming $T_{air}=0$ , finite element discretization of the heat transfer model leads to the parametrized system as below,

(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad y=C T,

where the volumetric heat capacity $\rho c_p$ , thermal conductivity $\kappa$ and the heat transfer coefficient $h$ between the membrane are kept as parameters. The volumetric hear capacity $\rho c_p$ is the product of two independent variables, i.e. the specific hear capacity $c_p$ and the density $\rho$ . The range of interest for the four independent variables are respectively $\kappa \in [2, 5]$ , $c_p \in [400, 750]$ , $\rho \in [3000,3200]$ , $h \in [10, 12]$ . The frequency range is $f \in [0,25]Hz$ . What is of interest is the output in time domain. The interesting time interval is $t \in [0,0.04]s$

Here $R(T)$ is either a constant heat resistivity $R(T)=R_0$ , or $R(T)=R_0(1+\alpha T)$ , which depends linearly on the temperature. Here we use $R_0=274.94 \Omega$ and temperature coefficient $\alpha=2.293 \pm 0.006 \times 10^{-4}$ . The model was created and meshed in ANSYS. It contains a constant load vector $B$ corresponding to the constant input power of $2.49mW$ . The number of degrees of freedom is $n=60,020$ .

The input function $u(t)$ is a step function with the value $1$ , which disappears at the time $0.02s$ . This means between $0s$ and $0.02s$ input is one and after that it is zero. However, be aware that $u(t)$ is just a factor with which the load vector B is multiplied and which corresponds to the heating power of $2.49mW$ . This means if one keeps $u(t)$ as suggested above, the device is heated with $2.49mW$ for the time length of 0.02s and after that the heating is turned off. If for whatever reason, one wants the heating power to be $5mW$ , then $u(t)$ has to be set equal to two, etc... When $R(T)=R_0(1+\alpha T)$ , it is a function of the state vector $T$ and hence, the system has non-linear input. (It is also called a weakly nonlinear system.)