Two-Dimensional Heat Transfer Considerations for Thermoreflectance Measurements

In the Suspended ThermoReflectance (STR) technique a microcantilever is heated with a laser power at the free end of the microcantilever and as heat propagates through it, another laser is used to measure the temperature along the beam.[1] In this paper, the heat equation is solved for two-dimensional heat flow in the microcantilever to determine the material’s thermal conductivity and heat capacity. Two of the dimensions of the microcantilever, width and length, are significantly greater than the third dimension, the thickness, leading to the two-dimensional approximation. Two boundaries along the length of the structure and one boundary along the width are assumed to be under Dirichlet boundary conditions, while the other boundary has Neumann condition. The Neumann or flux condition has a Gaussian profile due to the nature of laser beam intensity. The heat equation is solved using under 3 different flux conditions: (1) Steady-state, (2) Transient, and (3) Periodic. A steady-state condition mimics the experimental condition when a continuous wave laser is used to heat the microcantilever’s tip. A transient condition is possible when quickly removing or adding the continuous wave laser’s flux from the microcantilever’s tip using a chopper. Finally, a periodic condition can be achieved when an electro-optic modulator is utilized experimentally. Closed form analytical expressions are evaluated against the finite element model and experimental results for microcantilever beams and micro-structures of Si that have lengths on the order of a mm, width on the order of 100 microns, and thicknesses of 1 micron or less.