DOI: https://doi.org/10.30970/jps.01.241
A MODEL OF THE NARROW-BAND MATERIAL WITH THE ELECTRON–HOLE ASYMMETRY
|
Journal of Physical Studies 1(2), 241–250 (1997)
DOI: https://doi.org/10.30970/jps.01.241 A MODEL OF THE NARROW-BAND MATERIAL WITH THE ELECTRON–HOLE ASYMMETRY |
|
L. Didukh
Termopil State Ivan Puluj Technical University, 56 Rus'ka Str.,
UA-282001, Ternopil, Ukraine
E-mail: Yasniy@sci.politech.ternopil.ua
The Hamiltonian for describing the materials with a narrow energy band in the following form
\begin{eqnarray*}
H&=&{\sum}'_{ijσ}t_{ij}(n)α_{iσ}^{+}α_{jσ}+
U\sum_{i}n_{i\uparrow}n_{i\downarrow}+{\sum}'_{ijσ}
\left(J(iiij)α_{iσ}^{+}α_{jσ}n_{i\bar{σ}}+h.c.\right)
\\
&+&{1\over 2}{\sum}'_{ijσ σ^{'}}J(ijji)α_{iσ}^{+}
α_{jσ{'}}^{+}α_{iσ{'}}α_{jσ}+
{1\over2}{\sum}'_{ijσσ{'}} J(ijij)α_{iσ}^{+}α_{iσ}
α_{jσ{'}}^{+}α_{jσ{'}}
\end{eqnarray*}
is proposed; $α_{iσ}^{+},\ α_{iσ}$ are
operators of the creation and annihilation of electrons with the spin
$σ(σ=\uparrow,\downarrow)$ on $i$-site,
$n_{iσ}=α_{iσ}^{+}α_{iσ}$, $U$ is the intraatomic Coulomb
interaction, $n=
\begin{eqnarray*}
t_{ij}(n)=t(ij)+n~\sum_{{k\neq{i}},~{k\neq{j}}}J(ikjk)
\end{eqnarray*}
$t(ij),\ J(ijkl)$ are the matrix
elements of electron–ion and electron–electron interaction.
With the help of the perturbation theory an effective Hamiltonian is obtained taking into account the motion of the holes and double occupation states and indirect interactions. It is shown that in the doped Mott–Hubbards insulators cases $n<1$ and $n>1$ are nonequivalent (absence of electron–hole symmetry in contrast with the Hubbard model).
The insulating gap in case of $J(ijkl)=0,\ n=1$ is
\begin{eqnarray*} &&Δ E=-2w(1-2c)+(U^2+(4cw)^2)^{1/2}, \ \ \ \ w=z|t(ij)|, \ \ \ \ c=1/4+Uln(1-4c)/32cw\ \ \ \ (T=0,\ 2w>U) \end{eqnarray*}
and describes a transition from an insulating to a metallic state when the condition $2w>U$ is satisfied.
The results obtained are compared with some experimental data for narrow–band materials.