Silicon Nitride Membrane

Description of the device

A silicon nitride membrane (SiN membrane) 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 [1] and infrared sources [2].

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 $W m^{−3}$.

Description of the model

For the silicon nitride membrane under convection boundary condition and assuming the temperature of the air to be zero, can be rewritten as (C0 + ρcp · C1)˙T + (K0 + κ · K1 + h · K2)T = F U2(t) R(T ) y = E · T (11) where the volumetric heat capacity ρ ·cp, thermal conductivity κ and the heat transfer coefficient h between the membrane and the ambient air, are kept as parameters.

$\nabla \cdot (\kappa \nabla T) + Q- \rho c_p \frac{\partial T}{\partial t}=0, \quad Q=j^2R(T), \quad (1)$

where $\kappa(r)$ is the thermal conductivity in $W/(m*K)$ at the position $r, \, c_p$ is the specific heat capacity in $J /(kg*K), \, \rho(r)$ is the mass density in $kg /m^3,$ and $T(r,t)$ is the temperature distribution. We assume a homogeneous heat generation rate over a lumped resistor:

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

with unit $W/m^3$. 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 $W/(m^2 K)$.

Assuming $T_{air}=0$, spatial discretization of the heat transfer model in (1) leads to the parametrized system as below,

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

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 $\alpha=1.469 \times 10^{-3}$. The model was created and meshed in ANSYS. It contains a constant load vector 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.)

Data information

The system matrices are in MatrixMarket format (http://math.nist.gov/MatrixMarket/) and can be downloaded here File:Matrices gassensor.tgz. The files named by *.$A_i, \, i=0,1,2$ correspond to the system matrices $A_i, \, i=0,1,2$, respectively. The files named by $*.E_i, \, i=0,1,2$ correspond to $E_i, \, i=0,1$. The file named by $*.B$ corresponds to the load vector $B$ and the file named by $*.C$ corresponds to the output matrix $C$.

References

[1] T. Bechtold, "Model Order Reduction of Electro-Thermal MEMS", PhD thesis, Department of Microsystems Engineering, University of Freiburg, 2005.

[2] T. Bechtold, D. Hohfel, E. B. Rudnyi and M. Guenther, "Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction," J. Micromech. Microeng. 20(2010) 045030 (13pp).

Contact information:

Lihong Feng