Generalized WKB theory for electron tunneling in gapped $\alpha-\mathcal{T}_3$ lattices

TL;DR

This paper extends WKB semi-classical equations to analyze electron tunneling in generalized $ ext{alpha-} ext{T}_3$ lattices, revealing how transmission depends on energy gap, potential slope, and lattice geometry, with potential applications in ultrafast devices.

Contribution

It develops a generalized WKB framework for pseudospin-1 $ ext{alpha-} ext{T}_3$ materials with arbitrary hopping parameter, including new analytical solutions for electron wave functions.

Findings

Transmission depends on energy gap and potential slope.

Transmission amplitude strongly depends on lattice geometry-phase $ an^{-1} ext{alpha}$.

Results applicable to Dirac cone-based tunneling transistors.

Abstract

We generalize Wentzel-Kramers-Brillouin (WKB) semi-classical equations for pseudospin-1 $\alpha-\mathcal{T}_3$ materials with arbitrary hopping parameter $0 < \alpha < 1$, which includes the dice lattice and graphene as two limiting cases. In conjunction with a series-expansion method in powers of Planck constant $\hbar$, we acquired and solved a system of recurrent differential equations for semi-classical electron wave functions in $\alpha-\mathcal{T}_3$. Making use of these obtained wave functions, we analyzed the physics-related mechanism and quantified the transmission of pseudospin-1 Dirac electrons across non-rectangular potential barriers in $\alpha-\mathcal{T}_3$ materials with both zero and finite band gaps. Our studies reveal several unique features, including the way in which the electron transmission depends on the energy gap, the slope of the potential barrier profile and the transverse momentum of incoming electrons. Specifically, we have found a strong dependence of the obtained transmission amplitude on the geometry-phase $\phi = \tan^{-1} \alpha$ of $\alpha-\mathcal{T}_3$ lattices. We believe our current findings can be applied to Dirac cone-based tunneling transistors in ultrafast analog RF devices, as well as to tunneling-current control by a potential barrier through a one-dimensional array of scatters.