On the existence of free and self-trapped carriers in insulators: an abrupt temperature-dependent conductivity transition

Abstract

Abstract A variational calculation of the eigenstates of the three-dimensional analogue of Holstein's Molecular Crystal Model is utilized as a basis for determining the conditions under which carrier self-trapping does or does not occur in this system. It is found that below a temperature-dependent critical value of the electron-lattice coupling strength self-trapping does not occur; the eigenstates then correspond to an excess electron being only weakly coupled to the vibratory motion. Above a larger temperature-dependent critical value of the electron-lattice coupling strength only self-trapped (small-polaron) eigenstates exist. Between these two (temperature-dependent) critical values of the electron-lattice coupling strength both types of solutions are found. The condition for the existence of the weakly coupled situation, as well as that for the self-trapped circumstance, is shown to be derivable from arguments which are independent of the detailed variational calculation. These ancillary derivations provide a physical basis for understanding the two existence conditions. In addition, the results of the variational calculation are shown to agree with results obtained by non-variational means in the appropriate limits. The temperature-dependent appearance or disappearance of states from the energy spectrum of the coupled system manifests itself through an abrupt change in the electrical transport properties of the material. The conductivity, Hall mobility, and thermoelectric power on both sides of such a transition are calculated by an ad hoc application of previously obtained results. The essential feature of the occurrence of this transition is that the carriers on the low-conductivity side of the transition are self-trapped; they possess the low thermally activated mobility that is associated with small-polaron hopping motion. On the high-conductivity side of the transition the mobility of the carriers is considerably higher, it being that which is generally associated with electronic motion in rigid-lattice bands. The possible relevance of the present theory to the so-called insulator-to-metal and insulator-to-insulator transitions is discussed.