Exploring the graphene edges with coherent electron focusing

We study theoretically the coherent electron focusing in graphene nanoribbons. Using semiclassical and numerical tight-binding calculations we show that armchair edges give rise to equidistant peaks in the focusing spectrum. In the case of zigzag edges at low magnetic fields one can also observe focusing peaks but with increasing magnetic field a more complex interference structure emerges in the spectrum. This difference in the spectra can be observed even if the zigzag edge undergoes structural reconstruction. Therefore transverse electron focusing can help in the identification and characterization of the edge structure of graphene samples.

Figure 1

(Color online) Schematic geometry of the transverse-electron focusing setup. A graphene nanoribbon is contacted by an injector (I) and a collector (C) probe and perpendicular (to the graphene sheet) magnetic field is applied. Classical quasiparticle trajectories leaving from the injector at normal direction, depending on the magnetic field, can be focused onto the collector.

Figure 2

The results of tight-binding calculations for the transmission probability $T\left(B\right)$ as a function of magnetic field. (a) For an armchair; (b) for a zigzag nanoribbon. The black vertical lines indicate the positions of the focusing peaks predicted by the semiclassical theory, see Secs. 2, 3.

Figure 3

Comparison of the results for the band structure obtained from semiclassical quantization given by Eqs. (6, 7) (circles) and from numerical tight-binding calculations (solid lines). The energy is in units of $\hslash {\omega }_c=\sqrt{2}\frac{\hslash v_F}{l_B}$. The magnetic field is given by $W/l_B=8.97$. (a) Armchair nanoribbon; (b) zigzag nanoribbon, in the vicinity of the $\mathit{K}$ point.

Figure 4

The reczag edge. The numbers indicate the change in the hopping parameter on particular bonds with respect to its bulk value, $\gamma =-3.39\text{eV}$.

Figure 5

Comparison of the band structures of zigzag (solid line) and reczag (circles) nanoribbons from tight-binding calculations at the $K$ point. The strength of the magnetic field is given by $W/l_B=9.14$.