Calculation of Hubbard U Born Effect

The site charges are originally the same, eqnal to Z. After disproportionation, the charges (Z - 1) and (Z -i-1) alternate. The Bom radius R can be taken as the lattice constant. In atomic units, the Bom free energy (Section 6.2.2) difference between the spin-coupled state and the charged state is 

R is about 20 Bohr in fullerides. AG(Born) is thus equal to -1/20 Hartree or -1.3 eV. This lowers the energy difference between the spin-coupled state and the charged state, thus Hubbard U decreases. 

PbTe is a typical ionic crystal where Pb + alternates with Te . What happens if some of the Pb + are replaced by thallium. The Pb + sites are made for a +2 ion, but T1 does not want to be TI2+. It is found that half of the sites are TP and half of the sites are TP+. The 6s electrons form a unit electron pair and populate alternant thallium sites. The 6s singlet may transfer between sites as an electron pair, provided that there is near degeneracy between the charged state and the spin-coupled case. 

According to Equation 17.10 this comparison holds only for the low value of R that is relevant for TlTe. If the doping level is low, the distance R between the T1 sites is much larger. The Born contribution is now much smaller. The charged state is much less favored and near degeneracy is possible. Tl-doped PbTe is, in fact, superconducting below about 1 K in a certain. 

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