Comprehensive modeling of Joule heated cantilever probes
The thermo-electrical properties of a complex silicon cantilever structure used in thermal scanning probe lithography are modeled based on well established empirical laws for the thermal conductivity in silicon, the electrical conductivity in the degenerate silicon support structure, and a comprehensive physical model of the electrical conductivity in the low-doped heater structure. The model calculations are performed using a set of physically well defined material parameters and finite element methods to solve the coupled thermal and electrical diffusion equations in the cantilever. The material parameters are determined from a non-linear regression fit of the numerical results to corresponding measured data, which also includes Raman measurements of the heater temperature. Excellent agreement between predicted and measured data in the absence of air cooling is obtained if a tapered doping profile in the heater is used. The heat loss through the surrounding air is also studied in a parameter free three-dimensional simulation. The simulation reveals that the heater temperature can be accurately predicted from the electrical power supplied to the cantilever via a global scaling of the power in the power-temperature correlation function, which can be determined from the vacuum simulation.