Theoretical Analysis of a Dual-Probe Scanning Tunneling Microscope Setup on Graphene

Experimental advances allow for the inclusion of multiple probes to measure the transport properties of a sample surface. We develop a theory of dual-probe scanning tunneling microscopy using a Green’s function formalism, and apply it to graphene. Sampling the local conduction properties at finite length scales yields real space conductance maps which show anisotropy for pristine graphene systems and quantum interference effects in the presence of isolated impurities. Spectral signatures in the Fourier transforms of real space conductance maps include characteristics that can be related to different scattering processes. We compute the conductance maps of graphene systems with different edge geometries or height fluctuations to determine the effects of nonideal graphene samples on dual-probe measurements.

Figure 1

Schematic overview of a dual-probe STM setup. Current input and output probes and an impurity on site 0 are indicated together with their relative separations.

Figure 2

The conductance map for pristine graphene with $E_F=0.5|t|$. The fixed input probe is at the origin, and the map represents the conductance between the probes as a function of scanning probe position. The conductance has been multiplied by the interprobe distance $d_{12}$ to compensate for a geometric decay, see Eq. (3). The inset is a magnification of the boxed area.

Figure 3

Relative conductance map for $E_F=0.25|t|$ around a vacancy at (0, 0). The fixed probe (outside the scan area) at (0,106) nm is separated from the impurity along the armchair direction.

Figure 4

Fourier transform of the real-space map of $\mathrm{\Delta }\mathcal{T}$ for a single vacancy separated from the fixed probe along the armchair [(a) and (c)], and zigzag [(b) and (d)] directions. Energy is in the linear regime ($E=0.25|t|$) in (a) and (b), and beyond the linear regime ($E=0.5|t|$) in (c) and (d). (e) The Fermi surface of graphene beyond the linear regime. (f) and (g) Scattering diagrams for intra- and intervalley scattering.

Figure 5

(a) Illustration of dual-probe setup on a graphene edge. (b)–(c) Fourier transform of the conductance map near a zigzag edge (b) and an armchair edge (c) for $E_F=0.25|t|$. (d) Height profile of a nonplanar graphene sheet. (e) Conductance map for the nonplanar surface from (d) with the stationary probe at (0,0). (f) Transmissions along the dashed line in (b) and (e). See main text for further description.