Simple Model for Metal-Insulator Transition

In crystals, where some sites have unpaired electrons, these electrons may be localized or delocalized, depending on the influence of some external variable. If the system is localized, the electron has a certain residence time. The structure of a site with an electron is slightly different from a site without an electron. In Chapter 10, we learned that in this situation the reorganization energy X is larger than twice the coupling, equal to A. Suppose now that X is a parameter that may be varied, while A remains constant. (Alternatively we may keep X constant and vary A. ) 

The Hamiltonian matrix is diagonalized. The probability P,i for the electron being on site i has now decreased, since the ground state wave function coefficients, Cv, on the other sites are different from zcto. Thus, the remaining charge on site i after the first iteration is 

FIGURE 16.8 Metal-insulator transition, induced by varying X. 

If we go from the delocalized case and increase X, nothing happens for small X. After a further increase, self-trapping occurs on a certain site. 

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