Guided modes and terahertz transitions for two-dimensional Dirac fermions in a smooth double-well potential

R. R. Hartmann and M. E. Portnoi

Phys. Rev. A 102, 052229 – Published 30 November 2020

Abstract

The double-well problem for the two-dimensional Dirac equation is solved for a family of quasi-one-dimensional potentials in terms of confluent Heun functions. We demonstrate that for a double well separated by a barrier, both the energy-level splitting associated with the wave-function overlap of well states and the gap size of the avoided crossings associated with well and barrier state repulsion can be controlled via the parameters of the potential. The transitions between the two states comprising a doublet, as well as transitions across the pseudogaps are strongly allowed, highly anisotropic, and, for realistic graphene devices, can be tuned to fall within the highly desirable terahertz frequency range.

Received 27 August 2020
Revised 18 October 2020
Accepted 9 November 2020

DOI:https://doi.org/10.1103/PhysRevA.102.052229

Physics Subject Headings (PhySH)

Confinement Electro-optical spectra Quantum correlations, foundations & formalism Relativistic wave equations

1-dimensional systems 2-dimensional systems Graphene

General PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

R. R. Hartmann ^*

Physics Department, De La Salle University, 2401 Taft Avenue, 0922 Manila, Philippines

M. E. Portnoi ^†

Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom and ITMO University, St. Petersburg 197101, Russia

^*richard.hartmann@dlsu.edu.ph
^†M.E.Portnoi@exeter.ac.uk

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Vol. 102, Iss. 5 — November 2020

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Images

Figure 1
A schematic diagram of a double-well system, created by three carbon nanotube top gates. The central nanotube is negatively biased to create the central barrier, while the other two are positively biased to create the two wells. The Dirac material sits on top of a dielectric layer (violet layer), which lays on top of the metallic back gate. The Fermi level can be controlled using the back gate potential. The electrostatic potential created by the top gates is shown by the thick black line.
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Figure 2
Energy levels of 2D Dirac electrons in a double-well potential. The black solid lines show the two potentials given by Eq. (10), (a) for the case of $A_{1} = - 6$ and $A_{0} = 0$ and (b) for the case of $A_{1} = - 3$ and $A_{0} = - 2$ . The horizontal lines depict the bound-state energies for the four lowest doublet states, plus all the holelike states in the spectrum for the case of $M = 1$ . The red solid and blue dashed lines correspond to the eigenvalues of the even and odd modes of $ψ_{I}$ , respectively.
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Figure 3
The energy spectrum of bound states contained within a double well separated by a barrier, defined by the potential parameters (a) $A_{1} = - 6$ and $A_{0} = 0$ and (b) $A_{1} = - 3$ and $A_{0} = - 2$ , as a function of effective mass, $M$ . Here, only the four lowest doublet states are displayed and all barrier states are present within the energy range shown. The alternating red solid and blue dashed lines represent the even (odd) and odd (even) modes of $ψ_{I} (ψ_{I I})$ , respectively. The boundary at which the bound states merge with the continuum is denoted by the grey short dashed lines.
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Figure 4
The normalized bound-state wave functions of the double-well potential. $V = A_{1} \tilde{x} / (1 + {\tilde{x}}^{2})$ , where $\tilde{x} = | x^{'} | - 1, A_{1} = - 6$ , and $M = 3.5$ for the first three doublet states of energies: (a) $E = 0.92814$ , (b) $E = 0.92818$ , (c) $E = 1.54719$ , (d) $E = 1.54723$ , (e) $E = 1.98230$ , (f) $E = 1.98235$ , and for the hole states of energy (g) $E = - 1.10843$ , (h) $E = - 2.23465$ , and (i) $E = - 3.31137$ . The solid red and dash-dotted blue lines correspond to the even and odd modes of $Ψ_{I}$ , respectively, while the dashed and short dashed lines correspond to the other spinor component $Ψ_{I I}$ . The grey line shows the double-well potential as a guide to the eye, plotted in arbitrary units.
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Figure 5
The energy spectrum of bound states contained within a Rosen-Morse-double-well potential, defined by the potential parameters (a) $B_{1} = 14$ and $B_{2} = - 1$ , as a function of effective mass, $M$ . Here, only the four lowest doublet states are displayed, while all barrier states are present within the energy range shown. The alternating red solid and blue dashed lines represent the even (odd) and odd (even) modes of $ψ_{I} (ψ_{I I})$ , respectively. The boundary at which the bound states merge with the continuum is denoted by the grey short dashed lines.
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Figure 6
The energy spectrum for the three lowest-energy guided modes contained with a Lorentzian potential of strength $A_{0} = - 3.5$ , as a function of $M$ . The alternating red solid and blue dashed lines represent the even (odd) and odd (even) modes of $Ψ_{I}$ ( $Ψ_{I I}$ ), respectively. The black crosses denote the supercritical states. The boundary at which the bound states merge with the continuum is denoted by the grey short dashed lines.
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