  The reduction of the thermal conductivity in nanostructures opens up the
possibility of exploiting for thermoelectric purposes also materials such as
silicon, which are cheap, available and sustainable but with a high thermal
conductivity in their bulk form. The development of thermoelectric devices
based on these innovative materials requires reliable techniques for the
measurement of thermal conductivity on a nanometric scale. The approximations
introduced by conventional techniques for thermal conductivity measurements can
lead to unreliable results when applied to nanostructures, because heaters and
temperature sensors needed for the measurement cannot have a negligible size,
and therefore perturb the result. In this paper we focus on the 3$\omega$
technique, applied to the thermal conductivity measurement of suspended silicon
nanomembranes. To overcome the approximations introduced by conventional
analytical models used for the interpretation of the 3$\omega$ data, we propose
to use a numerical solution, performed by means of finite element modeling, of
the thermal and electrical transport equations. An excellent fit of the
experimental data will be presented, discussed, and compared with an analytical
model.
