Simulation of Thermal Positioning in Micro- and Nano-Scale Bridge Structures

Heat transfer in a thermally-positioned doubly-clamped bridge, at the micro- and nano-scale, is simulated to investigate the effect of convective cooling on the mechanical response of the system. The mechanical response of the system is defined as the displacement at the center of the bridge. The heat conduction equation is solved numerically using a finite difference method to obtain the temperature distribution in the bridge. Then, thermal stress due to the temperature difference with respect to the wall temperature is calculated. The thermo-structural equation is solved numerically to get the displacement along the beam. Two systems are compared: one doubly clamped beam with a length of 100 microns, a width of 10 microns, and a thickness of 3 microns, and a second beam with a length of 10 microns, a width of 1 micron, and a thickness of 300 nanometers, in air at a pressure from 0.01 Pa to 2 MPa. Conduction within the beam as well as convection between the beam and the gas are considered. A constant heat load with respect to the time is applied to the top of the beam varying from 10 to 600 μW/μm². The simulations use both free molecular and continuum models to define the convective coefficient, h. The simulations are performed for three different materials: silicon, silicon carbide, and diamond. The numerical results show that the displacement and the response of thermally-positioned nano-scale devices are strongly influenced by ambient cooling. The displacement depends on the material properties, the geometry of the beam, and the Biot number. In the free molecular model, the displacement varies significantly with the pressure at high Biot numbers, while it does not depend on cooling gas pressure in the continuum case. The significant variation of displacement starts at Biot number of 0.1 which occurs at gas pressure of 27 KPa in nano-scale. As the Biot number increases, the dimensionless displacement, δ* = δk/q^″αl² decreases. The displacement of the system increases significantly as the bridge length increases, while these variations are negligible when the bridge width and thickness change. Thermal noise analysis shows silicon carbide has the most physically meaningful displacements in comparison with silicon and cvd diamond.