Several approaches, using different forms of Green's functions, are presented for the two-dimensional analysis of slow-wave structures which are used in monolithic microwave integrated circuits. It is shown that these methods lead to an accurate determination of the depletion layer profile in the metal-semiconductor structure, allowing an adequate modelling of the corresponding cross-section. The most significant characteristics of a quasi-TEM slow-wave propagation, namely the slowing factor and attenuation constant, are calculated and in some cases the obtained theoretical results are shown to be in good agreement with available experimental data. It is also demonstrated that these methods allow to study single- and dual-gate travelling-wave field effect transistors by means of a simple change in the boundary conditions and that in this case, a "small-signal approach" conducts directly to the evaluation of complex inter-electrode impedances.