I carried out the study on the electronic states in semi-macroscopic systems, specially the transport phenomena.
1. I investigated the fundamental theory of transport phenomena in semi-macro system (mesoscopic system), and obtained the following results.
The most of the theoretical studies on the electrical conduction in mesoscopic system are based on the Landauer formula. In order that the Landauer formula be applicable, however, the are several implicit conditions to be fulfilled. If all of them are not fulfilled, we cannot expect that the Landauer formula gives a quantitatively correct results.
Naturally, the inelastic scattering of electrons by lattice vibration must not take place in the sample. Moreover, the scattering must not take place also in the boundary between the reservoir and the sample. In the boundary, there are electrical dipole layers, which give the potential difference between the reservoir and the sample. It is important that the scattering do not take place in these layers. On the contrary, the reservoirs must be large enough so that the electrons are fully thermalized in them. On such conditions, the electrons in each mode are completely incoherent to each other, and the current observed in the experiments are the simple sum of the contribution of the each mode. It is only under these condition that the double peak distribution, which has been observed in the experiment, is obtained.
The effects of the inelastic scattering between the electrons are very subtle and a definite conclusion is not yet obtained.
2. I investigated the quantization of the conductance of electrons through a narrow channel with numerical method on some realistic models, and I have clarified the conditions for the well define quantization.