Guiding Dirac Fermions in Graphene with a Carbon Nanotube

Abstract

Relativistic massless charged particles in a two-dimensional conductor can be guided by a onedimensional electrostatic potential, in an analogous manner to light guided by an optical fiber. We use a carbon nanotube to generate such a guiding potential in graphene and create a single mode electronic waveguide. The nanotube and graphene are separated by a few nanometers and can be controlled and measured independently. As we charge the nanotube, we observe the formation of a single guided mode in graphene that we detect using the same nanotube as a probe. This single electronic guided mode in graphene is sufficiently isolated from other electronic states of linear Dirac spectrum continuum, allowing the transmission of information with minimal distortion.Like a photon, an electron can be used as a carrier of information [1]. However, there is a limited number of tools to control a single electron [2] and the simple fact of guiding it coherently in a solid, like an optical fiber for light, is a technological feat [3, 4]. One-dimensional materials such as semiconducting nanowires naturally provide guidance for electrons, but in these materials, electrons can only be transmitted over short distances before losing its information [5]. Another possibility is through the edge channel of a two-dimensional electron gas in the quantum Hall regime, but a large magnetic field is required for the channel to be a single mode [6], which is crucial for the carried information not to be distorted during propagation.An alternative approach, conceptually similar to an optical fiber [7], is to use an electrostatic potential well on a two-dimensional electron gas to confine the movement of electrons along one direction ( Fig. 1a) [8][9][10][11]. Particularly, massless quasiparticles in graphene is an ideal platform for the realization of such electron guide. The quasirelativistic linear energy dispersion in graphene allows the wavefunction of the Dirac fermions travel with minimal distortion. Furthermore, it has been demonstrated that high mobility [12] allows electrons to be transmitted ballistically over several microns even at room temperature [13]. In addition, graphene can be encapsulated between thin dielectric layers of hexagonal-boron nitride (h-BN) [1], providing tunable electrostatic potential on the scale of a few nanometers, without degradation of the mobility. Electrostatic gating has produced various electron-optical elements, including lenses with negative refractive index [15] and filter-collimator switches [16].An ideal single mode electronic guide requires a deep potential well with a width much smaller than the wavelength of electrons in order to suppress scattering in the core of the waveguide [17]. The wavelength can reach around one hundred nanometers with experimentally accessible densities, and it is therefore crucial to be able to place extremely narrow gates close to the electron gas. The electronic modes generated by such a 1-dimensional -U 0 NT Au BN BN G 10µm 500nm CNT (a) (b) (c) charged CNT x y U(x,y) (d)...

Bipolar electron waveguides in graphene

“…The expression above can be easily modified when the top gates are fully embedded in a dielectric material. In the above-mentioned recent work [27],…”

“…Unlike a physical tube whose radius cannot be changed, externally applied potentials can be easily varied. There has been significant experimental progress since the pioneering work in the field of graphene electron waveguides [21][22][23][24][25][26] and the recent breakthrough of utilizing a nanotube as a top gates [27] enabled the detection of individual guided modes within a single waveguide. However, apart from graphene waveguides possessing a threshold in the characteristic potential strength required to observe a fully bound mode [14,28], one could argue that a single graphene electron waveguides provides similar physics to that studied in quasi-1D channels within conventional semiconductor systems.…”

Mapping borophene onto graphene: Quasi-exact solutions for guiding potentials in tilted Dirac cones

“…The conductance along such a channel can be measured by placing one terminal at each end. According to the Landauer formula, when the Fermi level is set to energy E (by modulating the back-gate voltage 21 ), the conductance along the waveguide is simply 2(n K + n K ′ )e 2 /h, where n K and n K ′ are the number of modes belonging to the s = 1 and s = −1 chirality, respectively (or the K and K ′ valley in the effective graphene sheet), at that particular energy. In a 2D Dirac material subject to a quasi-1D potential, the introduction of the tilt parameter breaks the E(∆) = E(−∆) symmetry for a given valley.…”

“…These exact and quasi-exact bound-state solutions have direct applications to electronic waveguides in 2D Dirac materials 2-5, 12, 13 , such as graphene, where the low-energy spectrum of the charge carriers can be described by a Dirac Hamiltonian 14 , and the guiding potential can be generated via a top gate [15][16][17][18][19][20] . Recent advances in device fabrication, utilizing carbon nanotubes as top gates, has enabled the detection of individual guided modes 21 , opening the door to several new classes of devices such as THz emitters 5,13 , transistors 12 , and ultrafast electronic switching devices 22 . These advances in electron waveguide fabrication technology make the need for analytic solutions all the more important, since they are highly useful in: determining device geometry, finding the threshold voltage required to observe a zero-energy mode 12 , calculating the size of the THz pseudo-gap in bipolar waveguides 13 , as well as ascertaining the optical selection rules 5,13 in graphene heterostructures.…”

Dirac fermion optics and directed emission from single- and bilayer graphene cavities

“…Correspondingly, optics analogues comprise Klein tunneling in single-layer graphene p-n-p junctions [1][2][3][4][5][6][7], p-n junctions [8][9][10], or Fabry-Pérot type settings [7,11,12] as well as anti-Klein tunneling in bilayer graphene [1,[13][14][15][16][17] where in particular circular p-n junctions were considered [18]. Collimation [8,19], various electron lensing [20][21][22][23], and guiding [24][25][26][27][28][29][30] phenomena were investigated in this context. Complementary to the use of top gates in several of the aforementioned electron steering experiments, recently a scanning tunneling setting has been employed to create disklike cavities in graphene defined by circular p-n junctions and to probe whispering-gallery-type resonant states that are most stable against decay from the cavity via Klein tunneling [31]; in a first subsequent theory work nonreciprocity of these whispering gallery modes was predicted [32].…”