Self-trapped magnetic polaron: Exact solution of a continuum model in one dimension

Abstract

A continuum model for the self-trapped magnetic polaron is formulated and solved in one dimension using a variational technique as well as the Euler-Lagrange method, in the limit of ${J}_{H}\ensuremath{\rightarrow}\ensuremath{\infty},$ where ${J}_{H}$ is the Hund's-rule coupling between the itinerant electron and the localized lattice spins treated as classical spins. The Euler-Lagrange equations are solved numerically. The magnetic polaron state is determined by a competition between the electron kinetic energy, characterized by the hopping integral t, and the energy of the antiferromagnetic lattice, characterized by the exchange integral J. In the broad-band case, i.e., for large values of $\ensuremath{\alpha}\ensuremath{\equiv}{t/JS}^{2},$ the electron nucleates a saturated ferromagnetic core region (type-II polaron) similar to the Mott description, while in the opposite limit, the ferromagnetic core is only partially saturated (type-I polaron), with the crossover being at ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}7.5.$ The magnetic polaron is found to be self-trapped for all values of $\ensuremath{\alpha}.$ The continuum results are also compared to the results for the discrete lattice.