Theoretical calculation of the dispersion relation for polymeric thin films: Determination of the thermal diffusivity using photothermal microscopy

Abstract

Photothermal reflectance microscopy can be used to characterize thin films grown on substrates in two different ways: (a) scanning modulation frequency with both the pump and probe beams coincident and (b) varying the pump–probe beam distance for several modulation frequencies. In the second case, the phase lag behavior, as a function of the beam separation, gives the effective wave number of the thermal wave along the surface. A plot of the wave number versus frequency provides a dispersion relation. Such a relation can be obtained by modeling the thermal problem of the system, and from the experimental data analysis one can determine the thermal parameters of the film and substrate. In the case of films with high thermal diffusivities, it is possible to find an analytical expression for the dispersion relation [A. A. Maznev et al., J. Appl. Phys. 78, 5266 (1995)]. However, in the case of films with low thermal diffusivities numerical calculation is needed. In this article we present the results of such a calculation for polymer films which have low diffusivities compared to the substrates (glass and silicon). The films were assumed to be opaque to both the pump and probe beams, i.e., both absorption and reflection occur at the film surface. It was observed that at high modulation frequencies the dispersion relation splits into two branches, one characteristic of the film, which is valid for small distances between the pump and probe, and the other characteristic of the substrate, valid for large distances.