Electronic analogue of Fourier optics with mass-less Dirac fermions scattered by quantum dot lattice

TL;DR

This work addresses realizing an electronic analogue of Fourier optics for massless Dirac fermions in graphene by scattering them from a gate-defined two-dimensional quantum dot lattice. It develops Fourier electron optics (FEO) through a Lippmann-Schwinger treatment, analyzes square, hexagonal, and moiré TDQDL configurations, and derives a Fraunhofer-like diffraction pattern alongside a Boltzmann-transport computation of angle-resolved resistivity. The key contributions include establishing a quantitative FEO dictionary, demonstrating a Babinet-like principle in electronic scattering, and showing that Fourier components of the resistivity reveal lattice symmetry, defects, and moiré geometry. The results point to potential electronic image-processing concepts and devices built on gate-defined QD lattices in graphene-based platforms.

Abstract

The field of electron optics exploits the analogy between the movement of electrons or charged quasiparticles, primarily in two-dimensional materials subjected to electric and magnetic (EM) fields and the propagation of electromagnetic waves in a dielectric medium with varied refractive index. We significantly extend this analogy by introducing an electronic analogue of Fourier optics dubbed as Fourier electron optics (FEO) with massless Dirac fermions (MDF), namely the charge carriers of single-layer graphene under ambient conditions, by considering their scattering from a two-dimensional quantum dot lattice (TDQDL) treated within Lippmann-Schwinger formalism. By considering the scattering of MDF from TDQDL with a defect region, as well as the moiré pattern of twisted TDQDLs, we establish an electronic analogue of Babinet's principle in optics. Exploiting the similarity of the resulting differential scattering cross-section with the Fraunhofer diffraction pattern, we construct a dictionary for such FEO. Subsequently, we evaluate the resistivity of such scattered MDF using the Boltzmann approach as a function of the angle made between the direction of propagation of these charge-carriers and the symmetry axis of the dot-lattice, and Fourier analyze them to show that the spatial frequency associated with the angle-resolved resistivity gets filtered according to the structural changes in the dot lattice, indicating wider applicability of FEO of MDF.

Figures (11)

• Figure 1: (a) The schematic diagram of a plane wave (direction shown in blue arrows) of charge carriers in ballistic graphene that are modelled as MDF under ambient conditions, getting scattered by a two-dimensional array of Gaussian quantum dot(QD) potentials created by STM tips. (b) The polar plot of the DSC for a square lattice of QDs as given by Eq. \ref{['square']}, of dimension $N_1=10$,$N_2=0$ and orientation $\phi=0$. The central maxima at $\theta=0$ is multiplied with $1.6\times 10^{-3}$ for better visibility. The first maxima on both sides are multiplied by $0.8\times 10^{-2}$. The second ones are multiplied by $0.32$ for better visibility with respect to the other smaller peaks. In the inset we have shown the differential scattering cross section for a single QD for differential values of $\beta$. In (c) and (d) the angle-resolved dc-resistivity of the system parallel to the direction of propagation of the incoming plane wave of graphene electrons is plotted under this scattering potential rotated at an arbitrary angle. The resistivity pattern for square and hexagonal lattices of QDs is shown in (c) and (d) for $N_2=100$. The resistivity at $\phi=0^{\circ}$ and $90^{\circ}$ is the same for the square lattice but not in the case of the hexagonal lattice. In Figs. (e)-(g) we compare the process described in (a)-(d) with the two-dimensional optical spatial frequency processor, whereas a short thesaurus listing various analogue quantities in these two systems is given in TABLE \ref{['table']}. In (e) we show that the object is positioned in the front focal plane of lens 1. The Fourier transform of the object distribution is found in the back focal plane of lens 1 as shown in (f). This plane is called spatial frequency plane Birch1968KGBirch_1972. At the image plane in (g) the object distribution is recovered.
• Figure 2: Resistivity pattern for (a) square QD lattice of size $N_2=200$ and $\Delta=1$ with square defect region with different sizes and (b) hexagonal QD lattice of size $N_2=61$ and $\Delta=1$ with circular defect region of different radiuses. In (a), The resistivity pattern is symmetric on both sides of $\phi=0$ only when the defect region is centred at the origin and for the blue curve, we have removed scatterers from $n_1=10$ to $110$ and $n_2=10$ to $110$. For the orange curve $n_1=10$ to $110$ and $n_2=90$ to $190$. In (b), the defect region is placed in the centre of the original QD lattice. Here, the resistivity pattern is symmetric on both sides of $\phi=0$.
• Figure 3: The schematic diagram of a plane wave of massless Dirac fermions getting scattered by a moiré superlattice of two (a) square and (b) hexagonal lattices of Gaussian quantum dots(QD) in graphene. Moiré pattern made by two square lattices of TDQDL producing a commensurate structure at a twist angle, $\delta \approx 6.026^{\circ}$ is shown in (c). The moiré lattice is shown in green, and the commensurate lattice is shown in black. Such quantum dots can be created in experimental system by using tips with applied gate voltage in the same way as in Fig. \ref{['figSL']}. In (d) $\abs{\Tilde{V}(\mathbf{q_1})}^2$ is plotted as a function of $q_{1x}$ and $q_{1y}$ for the above scattering potential. The resistivity pattern with fixed the mean angle($\phi$) is shown in (e) and (f) for a TDQDL scattering potential made with moiré pattern of two square and hexagonal lattices, respectively. In (g) and (h), the resistivity pattern is plotted with fixed twist angle($\delta$) again for a moiré pattern of two square and hexagonal lattices, respectively.
• Figure 4: (a) Shows the Fourier transform(FT) of the resistivity pattern for a TDQDL with $N_{2}=50$, $\Delta=1$ and $d=70$(nm). The blue cross ($\times$) denotes the value of amplitude corresponding to each spatial frequency ($l$). In the inset, we have shown the total data. The main figures do not show the central peak to display the smaller values. The FT of the resistivity pattern through a Gaussian filter for the same TDQDL scattering potential with a square defect region (in the centre) is shown in (b). In (c), we show the FT of the resistivity pattern through a Gaussian filter for a scattering potential made with a moiré pattern of two square TDQDL with the same lattice constant.
• Figure 5: Resistivity vs $N_{d2}$ plot for different values of $\phi$.