Anisotropic dynamics of charge carriers in graphene

Computer simulation by numerically solving the time-dependent Schrödinger equation was used to investigate the spreading of electronic wave packets on the graphene surface injected from a local probe. The simulations show a highly anisotropic in-plane dynamics following a ${60}^{\circ }$ angular periodicity even near the Fermi energy. The wave packet first tunnels onto the small graphene clusters below the tip and the electronic states of these clusters govern the further spreading of the electron on the graphene surface. It was found that in the vicinity of the injection point the molecular physical behavior dominates, but at larger distances the wave propagation is governed by solid-state physical rules. The calculations show complex charge-spreading phenomena at graphene grain boundaries. Our results reveal a new picture of charge propagation in graphene, which has important consequences for nanoelectronic applications.

Figure 1

Model geometry of the STM tip graphene system. (a) $-2.7$ eV equipotential surface of the potential. The surface is clipped at the boundary of the presentation box. (b) Grayscale image of the 2D cross section of the potential. The hyperbolic protrusion on the upper half plane and the horizontal row of black dots represent the vertical cross sections of the tip and graphene, respectively. Black (white) denotes smallest (largest) potential values. The dark gray level at the upper part is the negative potential inside the STM tip ($-9.81$ eV), the light gray level at the lower part is the vacuum potential (zero).

Figure 2

(a)–(c) Probability density on the graphene sheet for selected energies shown as color-coded 2D sections. We used a nonlinear color scale [see the scale bar in (c)]. Black corresponds to zero density, bright yellow to the maximum density. Each subimage is renormalized individually to its maximum density. In (a) and (b) the far region is shown on a separate (enhanced) color scale. Maximum density values are $2.33\times {10}^{-3}$ ($2.58\times {10}^{-5}$), $1.94\times {10}^{-3}$ ($3.60\times {10}^{-7}$), $4.52\times {10}^{-3}$ ($1.15\times {10}^{-3}$) for the near (far) regions of (a), (b), and (c), respectively. Probability density values are relative to that measured in the tip bulk. The graphene network is shown by thin orange lines. Size of the presentation window is 7.68 nm. The inset shows the P(E) probability on the graphene. (d)–(f) Angular distribution of the probability current for the three energy values shown in (a)–(c) in the far region. Magnitude is relative to the incoming probability current at that energy.

Figure 3

Selected snapshots from the time evolution of the probability density of wave packet shown as color-coded 2D sections. Black corresponds to zero probability. We used the same color scale on all images, determined by the maximum probability density on the graphene sheet. The graphene network is shown by thin orange lines. Time values are: 1.22 fs, 2.00 fs, 2.45 fs, 3.14 fs, 3.52 fs, 4.19 fs. See the text for details. Size of the presentation window is 3.84 nm in both $x$ and $y$ directions. The inset shows the probability current I(t) flowing from the tip into the graphene surface.

Figure 4

Frozen-and-cut $C_{6m^2}$ graphene clusters (see the text for details). (a) Energy level diagram. Those states with hexagonal symmetry are shown by large yellow circles. (b) Tunneling density of states for all PAH-like states with hexagonal symmetry in the [$-4,+4$] eV window. A broadening of 0.5 eV was applied, because of the small lifetime of these states. (c)–(g) Probability density for those states with hexagonal symmetry in the [$-4,+4$] eV window.

Figure 5

Angular distribution of the probability current density near the Fermi energy in the far region for different tip positions and different radial distances. (a), (c): Tip above an atom. (b), (d): Tip above the center of a C-C bond. The insets show the probability density distribution in the near region for the two tip positions.

Figure 6

Probability density on the graphene sheet with the grain boundary for selected energies shown as color-coded 2D sections. Black corresponds to zero density, bright yellow to the maximum density. The far region is shown on a separate (enhanced) color scale. The graphene network is shown by thin orange lines. Size of the presentation window is 7.68 nm.