Searching for confined modes in graphene channels: The variable phase method

Using the variable phase method, we reformulate the Dirac equation governing the charge carriers in graphene into a nonlinear first-order differential equation from which we can treat both confined-state problems in electron waveguides and above-barrier scattering problems for arbitrary-shaped potential barriers and wells, decaying at large distances. We show that this method agrees with a known analytic result for a hyperbolic secant potential and go on to investigate the nature of more experimentally realizable electron waveguides, showing that when the Fermi energy is set at the Dirac point, truly confined states are supported in pristine graphene. In contrast to exponentially decaying potentials, we discover that the threshold potential strength at which the first confined state appears is vanishingly small for potentials decaying at large distances as a power law; but nonetheless, further confined states are formed when the strength and spread of the potential reach a certain threshold.

Figure 1

Reflection coefficient vs the potential strength for zero-energy charge carriers incident on a hyperbolic secant potential of unit width, for several values of propagating wave vector along the waveguide. Confined states are indicated by the divergences (peaks). The plots are displaced vertically from each other for clarity.

Figure 2

(a) A schematic diagram of the experiment being considered. The top gate is charged and produces an electrostatic potential in the graphene plane. The top gate is treated as a wire of radius $r_0$ suspended above the graphene plane. The graphene layer is maintained at charge neutrality by charging the back gate to $V_{\mathrm{bg}}$. (b) Comparison of the realistic top-gate structure (dotted blue line) to the fitted hyperbolic secant model (solid red line) for $h_1/h_2=0.9$.

Figure 3

The appearances of bound states for the top-gate potential and the fitted exactly solvable hyperbolic secant waveguide, with $h_1/h_2=0.9$ and $d=1$. Boxes and crosses indicate propagating modes for the model and realistic top-gate potential correspondingly. The left panel shows the large-scale plot for $q_y\in [0,0.5]$. The right panel contains a logarithmic plot for small $q_y$ for the realistic potential only, indicating that the threshold for the power-decaying top-gate potential is vanishingly small.