## 2 Background: Electron Localization in Disordered Medium
### The Mott Transition  
Consider further, an electron bound to an ion in Cu or some other metal. As shown in Figure 9, as the screening length decreases, the bound states rise in energy. In a weak metal, in which the valence state is barely free, a reduction in the number of carriers (electrons) will increase the screening length, since  
\[r_{TF}\sim n^{-1/6}\,. \tag{6}\]  
This will extend the range of the potential, causing it to trap or bind more states-making the one free valance state bound.  
Now imagine that instead of a single defect, we have a concentrated system of such ions, and suppose that we decrease the density of carriers (i.e., in Si-based semiconductors, this is done by doping certain compensating dopants, or even by modulating the pressure). This will in turn, increase the screening length, causing some states that were free to become bound, leading to an abrupt transition from a metal to an insulator, and is believed to explain the metal-insulator transition in some transition-metal oxides, glasses, amorphous semiconductors, etc. This metal-insulator transition was first proposed by N. Mott and is called the Mott transition. More significantly Mott proposed a criterion based on the relevant electronic density such that this transition should occur [4; 111]. In Mott's criterion, a metal-insulator transition occurs when the potential generated by the addition of an ionic impurity binds an electronic state. If the state is bound, the impurity band is localized. If the  
Figure 9: Screened defect potentials. The screening length increases with decreasing electron density \(n\), causing states that were free to become bound.  
state is not bound, then the impurity band is extended. The critical value of \(\lambda=\lambda_{c}\) may be determined numerically [112] with \(\lambda_{c}/a_{0}\approx 1.19\), which yields the Mott criterion of  
\[2.8a_{0}\approx n_{c}^{-1/3}\,, \tag{7}\]  
where \(a_{0}\) is the Bohr radius. Even though electronic interactions are only incorporated in the extremely weak coupling limit, Thomas-Fermi Screening, Mott's criterion still works for moderately and strongly interacting systems [113].  
While the Mott and Anderson localization mechanisms are quite different, the TDOS can be used as an order parameter in both cases. In the Anderson metal-insulator transition, the transition is entirely due to disorder, with no interaction effects. In the Mott metal-insulator transition, although the described system is surely strongly disordered, these effects do not contribute to the mechanism of localization. Nevertheless, both transitions share the same order parameter. On the insulating side of the transition the localized states are discrete so that the typical DOS is zero, while on the extended side of the transition, these states mix and broaden into a band with a finite typical and average DOS. Therefore, both transitions are characterized by the vanishing typical DOS, thus it may serve as an order parameter in both cases.  
Finally, note that while the Mott transition is quite often associated with strong electronic correlations (in clean systems), for impurities in metals with screened Coulomb interactions, such transition occurs already in the weak coupling regime. Thus, any cluster solver which captures interaction effects, at least at the Thomas-Fermi level, (including DFT), with the additional condition to self-consist the impurity potentials, should be able to capture the physics of this transition.