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

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 $\hslash$, 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 $\varphi ={\tan }^{-1}\alpha$ 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.

15 pages, 6 figures

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• Dynamical optical conductivity for gapped $\alpha -{\mathcal{T}}_3$ materials with a curved "flat" band
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• Tunneling valley Hall effect induced by coherent geometric phase
• Superfluidity of dipolar excitons in a double layer of $\alpha -T_3$ with a mass term
• Electronic spectrum and optical properties of Y-shaped Kekule-patterned graphene: Band nesting resonance as an optical signature
• An experimental scheme for determining the Berry phase in two-dimensional quantum materials with a flat band
• Developing a semiclassical Wentzel-Kramers-Brillouin theory for $\alpha -{\mathcal{T}}_3$ model
• Transport properties of the pseudospin-3/2 Dirac-Weyl fermions in the double-barrier-modulated two-dimensional system
• Pseudospin-one particles in the time-periodic dice lattice: A new approach to transport control
• Enhanced Andreev Reflection in Flat-Band Systems: Wave Packet Dynamics, DC Transport and the Josephson Effect
• Coherent-scatterer enhancement and Klein-tunneling suppression by potential barriers in gapped graphene with chirality-time-reversal symmetry