Flat-Lens Focusing of Electron Beams in Graphene

Abstract

Coupling electron beams carrying information into electronic units is fundamental in microelectronics. This requires precision manipulation of electron beams through a coupler with a good focusing ability. In graphene, the focusing of wide electron beams has been successfully demonstrated by a circular p-n junction. However, it is not favorable for information coupling since the focal length is so small that the focal spot locates inside the circular gated region, rather than in the background region. Here, we demonstrate that an array of gate-defined quantum dots, which has gradually changing lattice spacing in the direction transverse to propagation, can focus electrons outside itself, providing a possibility to make a coupler in graphene. The focusing effect can be understood as due to the gradient change of effective refractive indices, which are defined by the local energy band in a periodic potential. The strong focusing can be achieved by suitably choosing the lattice gradient and the layer number in the incident direction, offering an effective solution to precision manipulation of electron beams with wide electron energy range and high angular tolerance.

A two-dimensional Dirac fermion microscope

“…As the lithographic resolution offered by high-end electron beam lithography can be of order 0.01 μm, there are nearly four orders of magnitude difference between the dimensions of the components and the characteristic transport lengths, such as the Fermi wavelength, which is λ F ≈2π 1/2 n −1/2 ≈35 nm at a carrier density of n =10 12 cm −2 , typical numbers for ballistic graphene devices. The 2D analogy of an electron gun can be realized by combining the functionality of basic components such as ballistic point contacts, apertures 47 , Veselago lenses 32 34 and superlattice collimators 50 51 . Focusing of Dirac fermions can be carried by p–n junctions, where the carrier density can be controlled by electrostatic gates independently in the p- and n-doped regions.…”

A domain-specific language and matrix-free stencil code for investigating electronic properties of Dirac and topological materials

“…Then the structure can be used as a coupler to other electronic units. Achieving this by tuning the gradient of the gate potential appears to be a very efficient way, which is more easily realized in practice than modifying the geometrical properties of the array such as the lattice gradient or the layer number (Tang et al, 2016).…”

“…As an example, we apply the Chebyshev time evolution scheme to the propagation and scattering of a Dirac electron wave packet on a graphene sheet with an imprinted gate-defined quantum dot array (Fehske et al, 2015; Pieper et al, 2013). This is a timely issue of of high experimental relevance (Caridad et al, 2016; Tang et al, 2016; Walls and Hadad, 2015). We mimic the quantum dot array by implementing the potential V n in (2) as…”

“…The situation becomes more complicated—but also more interesting—when the dot potentials are modulated spatially or energetically. In this case, a direction-dependent transmission (cascaded Mie scattering (Caridad et al, 2016)) or even the focusing of the electron beam outside the dots can be observed (Tang et al, 2016). Similar phenomena are demonstrated by Figure 9.…”

Electronic scattering, focusing, and resonance by a spherical barrier in Weyl semimetals

“…On the other hand, it is relevant to various problems of practical interests, especially as the advancing fabrication techniques of Weyl semimetals are paving the way for more controllable scientific explorations and possible device applications. Some related considerations first arose in the graphene system [17][18][19][20][21][22] and relevant experimental observations have been reported [23][24][25] .…”