Numerical simulation of the heat transfer in amorphous silicon nitride membrane-based microcalorimeters

Abstract

Numerical simulations of the two-dimensional (2D) heat flow in a membrane-based microcalorimeter have been performed. The steady-state isotherms and time-dependent heat flow have been calculated for a wide range of sample and membrane thermal conductivities and heat capacities. In the limit of high internal thermal conductivity and low membrane heat capacity, the sample heat capacity determined using the relaxation method with a single time constant is shown to be exact. The fractional contribution of the square 2D membrane border to the total heat capacity is calculated (∼24%). Analysis of the steady-state isotherms provide the 2D geometric factor (10.33) linking membrane thermal conductance to thermal conductivity, allowing extraction of the thermal conductivity of either the membrane itself or a sample deposited everywhere on the membrane. For smaller internal thermal conductivity and/or larger membrane heat capacity, systematic errors are introduced into the determination of heat capacity and thermal conductivity of a sample analyzed in the standard (single time constant) relaxation method, as has been previously shown for one dimension. These errors are due to both the changing contribution of the membrane border and to deviations from the ideal semiadiabatic approximation of the relaxation method. The errors are here calculated as a function of the ratios of thermal conductivity and heat capacity of sample and membrane. The differential method of measurement in which the sample heat capacity is taken as the difference between a relaxation method measurement with and without the sample is shown to give significantly smaller errors than the absolute errors of a single measurement. Under standard usage, high internal thermal conductivity is guaranteed by use of a thermal conduction layer such as Cu. The systematic error in this case is an underestimate of true sample heat capacity by less than 2%. The simulation was extended to thermal conditions where a single time constant relaxation approximation cannot be used, specifically, for a sample with low thermal conductivity. Because of the highly precise geometry of these micromachined devices, a comparison between measured and simulated steady-state and time-dependent temperatures is demonstrated to allow extraction of the heat capacity and thermal conductivity of this sample with less uncertainty due to elimination of the Cu heat capacity.