  In recent years, it has been pointed out that the maximum accelerating
gradient of a superconducting RF cavity can be increased by coating the inner
surface of the cavity with a multilayer thin-film structure consisting of
alternating insulating and superconducting layers. In this structure, the
principal parameter that limits the performance of the cavity is the critical
magnetic field or effective $H_{C1}$ at which vortices begin penetrating into
the superconductor layer. This is predicted to depend on the combination of the
film thickness. We made samples that have a NbN/SiO$_2$ thin-film structure on
a pure Nb substrate with several layers of NbN film deposited using DC
magnetron sputtering method. Here, we report the measurement results of
effective $H_{C1}$ of NbN/SiO$_2$(30 nm)/Nb multilayer samples with thicknesses
of NbN layers in the range from 50 nm to 800 nm by using the third-harmonic
voltage method. Experimental results show that an optimum thickness exists,
which increases the effective $H_{C1}$ by 23.8 %.
