Journal of Physical Studies 2(3), 362–370 (1998) DOI: https://doi.org/10.30970/jps.02.362 METAL–INSULATOR TRANSITION IN NARROW-BAND MODEL WITH NON-EQUIVALENT HUBBARD SUB-BANDS

L. Didukh, V. Hankevych, Yu. Dovhopyaty
Ivan Puluj Ternopil State Technical University, Chair of Physics,
56 Rus'ka Str., Ternopil, UA–282001, Ukraine
E–mail: didukh@tu.edu.te.ua

The transition from an insulating to a metallic state is studied in a model of narrow–band material with non–equivalent Hubbard sub bands at half filling and zero temperature. The model is defined by the on–site interaction $U$ and hopping integrals $t_{ij}=t(ij)+T_1(ij)$, $\tilde t_{ij}=t_{ij}+2T(ij)$ and $t'_{ij}=t_{ij}+ T(ij)$, where
\begin{eqnarray*}
&&T_{1}(ij)=\sum_{\stackrel{k\neq{i}}{k\neq{j}}}J(ikjk),\qquad T(ij)=J(iiij),
\end{eqnarray*}
$t(ij), J(ijkl)$ are the matrix elements of electron–ion and electron–electron interactions.

Single–particle Green's function is obtained by a variant of the method of an approximate second quantization in the generalized Hartree–Fock approximation. This quasiparticle Green's function depends on the concentration of the polar states $c$. Chemical potential of the electron system being
\begin{eqnarray*}
&&μ={(1-2c+2c^2)w-2c^2\tilde{w}\over (1-2c)(w+\tilde{w})}U,
\end{eqnarray*}
where $w$ — half–width of ‟hole" band, $\tilde{w}$ — half–width of doublon band. A dependence of the concentration of polar states $c$ on $U$, $t$, $\tilde{t}$, $t'$ is found. We obtain an energy gap as the function of $U$, $w$, $\tilde{w}$, $c$, and the hopping integral $t'$:
\begin{eqnarray*}
&&Δ E=-(1-2c)(w+\tilde{w})+{1\over 2}(Q_1+Q_2), \\
&&Q_1=\left\{ A(w-\tilde{w})^2-2BU(w-\tilde{w})+U^2+\left( t'\over t \right)^2(4cw)^2\right\}^{1\over 2}, \\
&&Q_2=\left\{ A(w-\tilde{w})^2+2BU(w-\tilde{w})+U^2+\left( t'\over \tilde{t}\right)^2(4c\tilde{w})^2\right\}^{1\over 2},\\
&&A=(1-2c)^2+8c^2(1-2c+2c^2), \quad B=1-2c+4c^2.
\end{eqnarray*}
The condition for a metallic state is determined:
\begin{eqnarray*}
w+\tilde{w}\geq U.
\end{eqnarray*}

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