Electronic geometric phase factors

The development of electronic geometric phase factors is governed by an adiabatic vector potential induced in the nuclear kinetic energy when we extend the Born-Oppenheimer separation to the degenerate pair of states [1, 23-25]. To see how the induced vector potential appears, we consider the family of transformations which diagonalize the excited state electronic coupling in the form... 

The presence of a conical degeneracy at q = 0 between the upper and lower linear Jahn-Teller surfaces of Eq. (2.9) accounts for the occurrence of nontrivial electronic geometric phase factors in this system [1,2, 26],... 

The geometric phase effect associated with chemical reactions [166] and with the motion of electrons in magnetic fields [167] was generalized in 1984 by Berry [168] to systems which are transported around a loop or circuit C in parameter space. If H is the system s Hamiltonian and R a set of parametric variables on which H depends, he showed that an eigenstate of when the system is transported slowly (i.e., adiabatically) around C, will acquire a geometrical phase factor independent of time, in addition to the familiar... 

Mead and Truhlar [52] introduced an elegant way of incorporating the geometric phase effect, namely the vector potential approach. In this method, the real electronic wave function 4>(a), where a is any internal angular coordinate describing the motion around the Cl, is multiplied by a complex phase factor c(a) to ensure the single-valuedness of the new complex electronic wave function ... 

Unlike a geometrical factor, the value of the factor

with composition in a predictable way. To illustrate this, suppose that stoichiometric MO2 is heated in a vacuum so that it loses oxygen. Initially, all cations are in the M4+ state, and we expect the material to be an insulator. Removal of O2- to the gas phase as oxygen causes electrons to be left in the crystal, which will be localized on cation sites to produce some M3+ cations. The oxide now has a few M3+ cations in the M4+ matrix, and thermal energy should allow electrons to hop from M3+ to M4+. Thus, the oxide should be an n-type semiconductor. The conductivity increases until

reduction continues, eventually almost all the ions will be in the M3+ state and only a few M4+ cations will remain. In this condition it is convenient to imagine holes hopping from site to site and the material will be a p-typc semiconductor. Eventually at x = 1.5, all cations will be in the M3+ state and M2C>3 is an insulator (Fig. 7.3). 

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