Energy spectrum of the bound polaron.

Abstract

An eigenvalue problem for an electron interacting with a Coulomb center and a field of LO phonons is solved by a method of optimized canonical transformation. This method can be applied to arbitrary values of the electron-phonon coupling constant \ensuremath{\alpha}. The energy eigenvalues for the 1s through 4f states have been calculated as function of \ensuremath{\alpha} and of the ratio R of the donor rydberg ${m}_{e}$${e}^{4}$/2${\ensuremath{\Elzxh}}^{2}$${\ensuremath{\epsilon}}_{0}^{2}$ to the LO-phonon energy \ensuremath{\Elzxh}\ensuremath{\omega}. These values are the upper bounds to the energy ${E}_{1s}$ of the ground state as well to all the energy levels of the excited states lying below ${E}_{1s}$+\ensuremath{\Elzxh}\ensuremath{\omega}. In a broad range of \ensuremath{\alpha} and R, the present upper bounds are lower than previous variational results for the states 1s, 2s, and 2p. The energy levels for the 3s--4f states have been calculated for the first time by variational means. The calculated energy eigenvalues ${E}_{\mathrm{nl}}$ lie always below the corresponding hydrogenlike levels, i.e., ${E}_{\mathrm{nl}}$/\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\le}-\ensuremath{\alpha}-R/${n}^{2}$, where n and l are the principal and angular momentum quantum numbers, respectively. For all values of \ensuremath{\alpha} and R, the following sequence of the energy levels for a given n has been obtained: ${E}_{\mathrm{nl}}$\ensuremath{\le}${E}_{\mathrm{nl}\mathcal{'}}$ if l>l'. In particular, it leads to the positive Lamb shift ${E}_{2s}$-${E}_{2p}$. The model of the bound polaron has been applied to the description of shallow donor spectra. The calculated values agree rather well with the measured 1s-2p transition energies for CdTe and ZnSe, and 1s-2s transition energies for CdS. For AgBr, AgCl, and ${\mathrm{CdF}}_{2}$ the upper bounds for the 1s level are too low, but the 2p-3p energy differences agree well with the experimental data. It means that the short-range donor potential neglected in the polaron model is repulsive for the considered impurities in the ionic crystals.