  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.
