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This work analyzes the Abrikosov's idea that high temperature conductivity is based on resonant tunneling. A rectangular rod was considered and it was shown that high values of transparency coefficient can be produced in it. It is the consequence of the fact that electrons as well as phonons have energy gap within a rectangular rod.

The Hamiltonian of a quantum rod with an ellipsoidal boundary is given after a coordinate transformation, which changes the ellipsoidal boundary into a spherical one. We study the electron which is strongly coupled to the LO-phonon eigenenergies and eigenfunctions of the ground and the first-excited states in a quantum rod under an applied electric field by using variational method of Pekar type. This quantum rod system may be used as a two-level qubit. When the electron is in the superposition state of the ground and the first-excited states, we obtain the time evolution of the electron probability density. The probability density of the electron oscillates in the quantum rod with an oscillation period. It is found that due to the presence of the three-dimensional anisotropic harmonic potential in the radius and the length directions of the quantum rod, the electron probability density shows double-peak configuration, whereas there is only peak if the confinement is a two-dimensional symmetric one in the x- and y-directions. The oscillation period is an increasing function of the ellipsoid aspect ratio and the transverse and longitudinal effective confinement lengths of the quantum rod, whereas it is decreasing one of the electron–phonon coupling strength and the electric field.

The vibrational frequency and the ground-state energy of strong-coupling polaron in quantum rod (QR), bounded in parabolic potential with ellipsoidal boundary condition, are respectively, obtained using the linear-combination operator and unitary transformation methods. Numerical results illustrate that the vibrational frequency will increase with decreasing the effective radii R₀ of ellipsoidal parabolic potential and the aspect ratio e' of ellipsoid while with increasing electron-bulk longitudinal-optical (LO)-phonon coupling strength α, and that the ground-state energy will increase with decreasing R₀ and α. Besides, the ground-state energy will decrease with e' increasing in 0 < e' < 1 area, get to minimum when e' = 1, and then increase with e' increasing in e' > 1 area.