Analytical consideration of particle transport in 1D nanostructures

Abstract

The paper presents an analytical study of one-dimensional fluxes of ballistic quasiparticles in the presence of scattering centers. Such a situation can be realized at very low temperatures or systems of very small sizes – nanostructures. To describe such a situation, the approach of heat transfer by radiation, which goes back to Casimir, is used, in which the interaction of phonons with image boundaries is taken into account, or, for example, the Landauer approach, where the probability of phonon transition from the initial state to the final state is introduced. At the same time, the intermediate regime, the mean free path of phonons due to their interaction with each other, is comparable to the size of the samples; to this day, it remains a rather difficult problem for a theoretical or numerical solution. In this work, we propose the probabilistic approach in the Landauer model to describe heat transfer in the one-dimensional ballistic motion of quasiparticles. Within the framework of the theory of random walks, a model of successive scattering centers is considered. An explicit analytical expression is obtained for the dependence of the flux of quasiparticles on the probability of scattering and the number of scattering centers. In order to explain the physical sense of the obtained result the comparison with the result of iterative approach is made. As well the results are used for description of the problem of the heat flux in multilayered structures, in which one should take into account not only the thermal resistance inside the layers, but also the Kapitsa resistance between the layers. The practical application of the obtained results to one-dimensional nanostructures and to quasi-one-dimensional heat-conducting systems is discussed, various limiting cases are considered and a comparison with experimental data is made.