Polarons in an ellipsoidal potential well

Abstract

Applying the Feynman variational principle, we analyze the basic polaron parameters – the ground-state energy, the polaron effective mass, the number of phonons in the polaron cloud, and the polaron radius – for a three-axis ellipsoidal potential well. Numerical calculations are performed for the specific cases when the potential well determines (i) a cylindrical quantum wire, (ii) a planar quantum wire, and (iii) a spherical quantum dot. The polaron parameters are derived analytically for the limiting cases of large and small cross-section sizes of a quantum wire, and of large and small radii of a quantum dot. A boundary between the weak and strong coupling regions is studied as a function of sizes of the structures under consideration. It is shown that with confinement strengthening, regions of the weak and intermediate coupling shorten, while the strong-coupling region widens. For a “squeezed” polaron state in a quantum dot, when the radius of confinement R is smaller than the polaron radius R p =(ℏ/ m ̄ ω 0) 1/2 (with the electron band mass m ̄ and the LO phonon frequency ω 0), the electron–phonon coupling constant α is scaled as αR p/ R.