Temperature distribution in membrane-type micro-hot-plates with circular geometry

The absence of an accurate analytical model for the temperature distribution in micro-hot-plates does not allow the systematic design of micro-heaters with high temperature uniformity in the hot region, a key issue for the fabrication of accurate gas sensors, infrared emitters with high spectral purity, and micro-reactors with uniform deposition on sufficiently large areas. Here, by considering a circular heater geometry and typical (i.e. very small) thicknesses, we reduce the three-dimensional temperature distribution to one dimension only and, by solving a form of Bessel differential equation, we express the temperature distribution in terms of modified Bessel functions. The resulting relations accurately approximate the radiation heat transfer within the heater, which is a decisive advantage as the temperature within the heater is generally very high. Finally, we demonstrate that our model has almost the same accuracy as finite element method (FEM) simulations and is therefore suitable for designing micro-hot-plates with unprecedented temperature homogeneity as well as for accurately predicting the temperature profile in generic micro-hot-plates.