  The Casimir effect is a quantum phenomenon induced by the zero-point energy
of relativistic fields confined in a finite-size system. This effect for photon
fields has been studied for a long time, while the realization of counterparts
for fermion fields in Dirac/Weyl semimetals is an open question. We
theoretically demonstrate the typical properties of the Casimir effect for
relativistic electron fields in Dirac/Weyl semimetals and show the results from
an effective Hamiltonian for realistic materials such as Cd$_3$As$_2$ and
Na$_3$Bi. We find an oscillation of the Casimir energy as a function of the
thickness of the thin film, which stems from the existence of Dirac/Weyl nodes
in momentum space. Experimentally, such an effect can be observed in thin films
of semimetals, where the thickness dependence of thermodynamic quantities is
affected by the Casimir energy.
