A new type of neutron detector, named Stack Structure Solid organic Scintillator (S$^4$), consisting of multi-layer plastic scintillators with capability to suppress low-energy $gamma$ rays under high-counting rate has been constructed and tested. To achieve $it{n}$-$gamma$ discrimination, we exploit the difference in the ranges of the secondary charged particles produced by the interactions of neutrons and $gamma$ rays in the scintillator material. The thickness of a plastic scintillator layer was determined based on the results of Monte Carlo simulations using the Geant4 toolkit. With layer thicknesses of 5 mm, we have achieved a good separation between neutrons and $gamma$ rays at 5 MeV$_{rm ee}$ threshold setting. We have also determined the detection efficiencies using monoenergetic neutrons at two energies produced by the $it{d}$+$it{d}toit{n}$+$^{3}$He reaction. The results agree well with the Geant4 simulations implementing the Li$grave{rm e}$ge Intranuclear Cascade hadronic model (INCL++) and the high-precision model of low-energy neutron interactions (NeutronHP).
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