Nanoscale physics

The energy shift of particles in the quantum dot caused by their interaction with polarizing phonons
Tovstyuk C.C.
Department of Electronics and Information Technology, Lviv Polytechnic National University. Profesorska str., 2, Lviv-79013, Ukraine. 
E-mail: cctovstyuk@gmail.com

In this paper we analyze the effect of electron-phonon interaction on the energy of a particle in a quantum dot with one bound state (energy level). The energy (frequency) of phonons is supposed to be constant (not depending on quasi-momentum in the Brillouin zone). The results of our consideration can be used for the determination of the impurity energy shifts in semiconductors, or electrons in the Van der Waals slit of layered crystals. Moreover, these results can be also applied for systems of quantum dots in other environments.
The renormalized energy is determined by the poles of the Fourier image of Greens’ function, which can be represented by the Dyson equation. The energy shift is determined by the mass operator. The representation of a mass operator in a series on the parameter of electron-phonon interaction can be depicted by Feynman diagrams, or described by double permutations proposed in [1]. For the case when the dispersion of optical phonons can be neglected, considering their energy (frequency) to be constant in all Brillouin zone, this series will be greatly simplified due to the fact that analytical expressions corresponding to topologically different diagrams are equal. So the direct calculation of the mass operator for some (first, second and third) orders of approximations can be easily evaluated and the system response on one-, two- and three-phonon interaction can be obtained.
In this paper we show how the convolution of Feynman diagrams caused by non-dispersion one-particle energies reflects in the formality of double permutations.
The resulting expression for mass operator can be represented by infinite chain fractions, thus we investigated the convergence of such a series using the Vorpitski’ theorem. We determined the shift of energy level for different energies in the quantum dots (which are almost influent by the dot sizes) and for different constants of electron-phonon interaction (typical for the surrounding dot materials). The obtained results are in good agreement with known experimental data. 

1.	Tovstyuk C.C. Topologically distinct Feynman diagrams for mass operator in electron – phonon interaction // Condens. Matter Phys.-2009.-12.-P. 225-239.