Dual-probe spectroscopic fingerprints of defects in graphene

Recent advances in experimental techniques emphasize the usefulness of multiple scanning probe techniques when analyzing nanoscale samples. Here, we analyze theoretically dual-probe setups with probe separations in the nanometer range, i.e., in a regime where quantum coherence effects can be observed at low temperatures. In a dual-probe setup the electrons are injected at one probe and collected at the other. The measured conductance reflects the local transport properties on the nanoscale, thereby yielding information complementary to that obtained with a standard one-probe setup (the local density of states). In this work we develop a real-space Green's function method to compute the conductance. This requires an extension of the standard calculation schemes, which typically address a finite sample between the probes. In contrast, the developed method makes no assumption of the sample size (e.g., an extended graphene sheet). Applying this method, we study the transport anisotropies in pristine graphene sheets, and analyze the spectroscopic fingerprints arising from quantum interference around single-site defects, such as vacancies and adatoms. Furthermore, we demonstrate that the dual-probe setup is a useful tool for characterizing the electronic transport properties of extended defects or designed nanostructures. In particular, we show that nanoscale perforations, or antidots, in a graphene sheet display Fano-type resonances with a strong dependence on the edge geometry of the perforation.

Figure 1

Setup sketch including the leads modeled as one-dimensional chains with a hopping $t_c$ between the sites. The surface Green's functions $g_s$ are indicated together with the coupling ${\gamma }^{1/2}$ between the lead and the graphene sample.

Figure 2

(a) Sketch showing the pristine sample and the rotation angle $\theta$ from the armchair direction. (b)–(d) The transmission as a function of energy between the two leads separated by 50 nm along (b) armchair, (c) zigzag, and (d) rotated $\theta =11.1^{\circ }$ from the armchair direction. In (b) and (c) the transmission calculated using the SPA is indicated (red dots). (e) The oscillation period $\nu \left(\theta \right)$ (see main text for definition) is plotted against rotation angle $\theta$ as defined in (a). The curve is constructed by averaging over many individual calculations with distances ranging from 20 to 100 nm.

Figure 3

The transmission as a function of energy for pristine graphene (dashed), vacancy (red), and adatom (blue). The impurity is in between probes, which are separated by $\sim$50 nm along the armchair direction. The dots denote a similar calculation using the SPA expression Eq. (13). The parameters for the adatom are chosen as in Ref. [56] as ${\epsilon }_{\alpha }=-0.185\left|t\right|$ and $t_{\alpha }=0.37\left|t\right|$.

Figure 4

(a) Sketch illustrating the two probes separated along the armchair direction by $\sim$50 nm. The marks refers to impurity positions. Blue is along the line of separation and equidistant of the probes. The green and red squares are moved relative to the blue site along the armchair direction (parallel) by 12.8 nm and 34 nm, respectively. The transmission for the parallel translation is shown in (b) and (c). The green and red circles are equidistant of the probes but moved along the zigzag direction (perpendicular) to 7.4 nm and 17.2 nm, respectively. The transmission function for impurities in these positions is shown in (d) and (e). The zero point for the curves has been translated for better distinction between curves.

Figure 5

(a) Configuration-averaged transmission as a function of energy. (b) The difference between the averaged transmission and the pristine transmission. We place impurities in a $50\times 85$ nm square around the probes The unequal sides are chosen to take into account the probe separation direction. The curves are made from averaging $2\times {10}^4$ configurations.

Figure 6

The density of states for $E=0.028\left|t\right|$ around antidots with different edge structures as indicated. The maps are individually scaled.

Figure 7

(a) The transmission for probes separated along the armchair direction $(\sim 50$ nm) for zigzag, armchair, and circular antidots, respectively. The antidot structures are shown in Fig. 6. (b) Transmission for the same zigzag antidot as (a) including disorder of varying strength. Each curve is an average of 50 different configurations and has been shifted relative to the others. (c) The transmission for the same zigzag antidot as (a), with probe separations $(\sim 50$ nm) along armchair and zigzag directions, respectively. (d) Single-probe spectroscopy of zigzag antidot with the same probe position as (a). Calculation both with and without disorder is included. The curves have been shifted relative to each other.