  Recently, new quantum effects have been studied in thin nano-grating layers.
Nano-grating on the surface imposes additional boundary conditions on the
electron wave function and reduces the density of states (DOS). When the
nano-grating dimensions are close to the de Broglie wavelength, the DOS
reduction is considerable and leads to changes in the layer properties. DOS
calculations are challenging to perform and are related to the quantum billiard
problem. Performing such calculations requires finding the solutions for the
time-independent Schrodinger equation with Dirichlet boundary conditions. Here,
we use a numerical method, namely the method of auxiliary sources, which offers
significant computational cost reduction relative to other numerical methods.
We found the first five eigenfunctions for the nano-grating layer and compared
them with the corresponding eigenfunctions for a plain layer by calculating the
correlation coefficients. Furthermore, the numerical data were used to analyze
the DOS reduction. The nano-grating is shown to reduce the probability of
occupation of a particular quantum state, reducing the integrated DOS by as
much as 4.1 -fold. This reduction in the DOS leads to considerable changes in
the electronic properties.
