Valence transition behavior of the doped Falicov-Kimball model at nonzero temperatures

The extrapolation of small-cluster exact-diagonalization calculations is used to study the influence of doping on valence transitions in the spinless Falicov-Kimball model at nonzero temperatures. Two types of doping are examined, and namely, the substitution of rare-earth ions by non-magnetic ions that introduce (i) one or (ii) none additional electron (per non-magnetic ion) into the conduction band. It is found that the first type of substitution increases the average $f$-state occupancy of rare-earth ions, whereas the second type of substitution has the opposite effect. The results obtained are used to describe valence transition behavior of samarium in the hexaboride solid solutions $Sm_{1-x}M_xB_6$ ($M=Y^{3+},La^{3+}, Sr^{2+},Yb^{2+}$) and a very good agreement of theoretical and experimental results is found.