master : Vitali Ghoghoberidze Supervisor : Full Professor Avtandil Tavkhelidze PhD David Kakulia

 Introduction  thermal electricity  electromagnetic equations as analog quantum billiards  M.A.S  Computer modeling of new type of Thin films.  Conclusion

 It is well known that physical and chemical properties of nano structure Depends on the their dimension. besides this, the properties dependes on the geometry. For examples: carbon nanotube and graphene periodic structure changes The properties of Semiconducting material. This is because that periodic structure Affixed the additional boundary condition to the electrons wave function. Additional boundary conditions forbit some quantum states, so electron has to Occupy the QUANTUM leveles with higher energy. So the chemichal potential increase

 New materials are required for a new generation of energy converters and coolers. Nanostructuring allows changing material properties so that they satisfy the requirements of a particular energy conversion device. High thermoelectric figure of merit ZT materials is essential for a new generation of thermoelectric converters and coolers. Here, T is the temperature and Z is given by, Z=σ S 2 /(k e +k 1 )  The difficulty in increasing ZT is that materials having high S usually have σ low. When σ is increased, it leads to an increase in k e, following Wiedemann– Franz law, and ZT do not improve much. NG allows increasing of S without σ changing. NG layers can serve as high ZT materials for a new generation of thermoelectric converters and coolers.

Nanograting changes material properties. These changes are fully determined by the density of quantum states ρ(E). Changes in itself can be described by introducing geometry factor G such that ρ(E)= ρ0(E)/ G where, ρ0(E) is density of states in plain layer. To calculate G Schrodinger time-independent equation should be solved in Nanograting geometry. For most geometries, separation of variables is not possible and analytical solution does not exist. It was one of the goals of the work to solve this problem using numerical methods. There is a full mathematical analogy between quantum billiards and electromagnetic resonators. Therefore, it is reasonable to use the Method of Auxiliary Sources (MAS) for quantum billiard calculation, as it is most efficient numerical approach for solving eigen value problems

 This method has been propoused by Georgian mathematician V. Kupradze in It is widely used for solving various electromagnetic problems. Later on, the Method was adopted by Georgian scientists for solving eigen value problems related to wave guides with arbitrary cross-section. In the MAS, field is represented by fields of Helmholtz equation fundamental solutions (auxiliary sources), which radiation centers are placed on some auxiliary contour. Amplitudes of auxiliary sources can be found by satisfaction of boundary conditions. Near eigen values field increases inside of the area, that allow as to find them. method of auxiliary sources Eigen value problem for Laplace operator s

We wrote our program in MATLAB code. it contains two files. First part generated digital array analogous to the boundary conditions of quantum billiard and creates 100 auxiliary sources on its basis. The second part of the program sums radiation from the sources and finds wave vector value for which radiation is maximal inside the quantum billiard and equals to zero on its border first resonance in nanograting quantum billiard Second resonance in quantum billiard Contour

Spectrum of nanograting billiard

 For calibration we examined Square Geometry and calculated Resonance by the MAS using the MatLab program and found the results match with well known analytic results. k

 The Method of Auxiliary Sources is applied to solve quantum billiard problem for arbitrary shaped contour, based on full mathematical analogy between electromagnetic and quantum problems.  MATLAB code is developed based on the MAS, Computer simulations are performed.  Obtained numerical results are in a good argument with theoretical data.