Regulation of Electronic Structure in Armchair $\alpha-\mathrm{T}_{3}$ Nanoribbons Driven by Magnetic Filed

The
$\alpha-\mathrm{T}_{3}$
lattice is an intermediate model between the honeycomb lattice and the dice lattice (or T
3
lattice). Due to the characteristics of its lattice structure, a flat band in its energy band structure is stable, which has an attracted great attention among researchers. However, studies on
$\alpha-\mathrm{T}_{3}$
nanoribbon and the regulation of its bands with magnetic fields have rarely involved. Here we theoretically study the regulation of different magnetic fields on different nanoribbon widths, and the former's influence on the latter's transport properties. We take the armchair edged
$\alpha-\mathrm{T}_{3}$
nanoribbon (
$\mathrm{A}\alpha\mathrm{T}_{3}\text{NR}$
) with the widths
$N=18$
, 19, and 20 respectively for research. Based on the tight-binding model, adopting Bloch's theorem, and using the Green's function method, we have studied the band structure characteristics and transport properties. Through the research we found that no matter what the value of
$\alpha$
is, the
$\mathrm{A}\alpha\mathrm{T}_{3}\text{NR}$
has an complete FB at E
F
and its width meets the
$3p$
-conducting rule. For example
$N=3p+2$
are metallic (where
$p$
is a positive integer). Interestingly, the magnetic field can control the properties of the nanoribbon, allowing the metal band to open the gaps, but it has no effect on the flat bands at the Fermi level. The magnetic field changes continuously from 0 to 1, the gap adjustment of the nanoribbon is periodic, leading to a transition from metallic to semiconducting. It is believed that the findings have important applications for future magnetic control switches of
$\alpha-\mathrm{T}_{3}$
nanoribbon-based devices.