Characterizing wave functions in graphene nanodevices: Electronic transport through ultrashort graphene constrictions on a boron nitride substrate

We present electronic transport measurements through short and narrow $\left(30\times 30\mathrm{nm}\right)$ single-layer graphene constrictions on a hexagonal boron nitride substrate. While the general observation of Coulomb blockade is compatible with earlier work, the details are not: We show that the area on which charge is localized can be significantly larger than the area of the constriction, suggesting that the localized states responsible for the Coulomb blockade leak out into the graphene bulk. The high bulk mobility of graphene on hexagonal boron nitride, however, seems to be inconsistent with the short bulk localization length required to see Coulomb blockade. To explain these findings, charge must instead be primarily localized along the imperfect edges of the devices and extend along the edge outside of the constriction. In order to better understand the mechanisms, we compare the experimental findings with tight-binding simulations of such constrictions with disordered edges. Finally, we discuss previous experiments in the light of our findings.

Figure 1

(a) Scanning force microscopy image (false color) of the two devices A and C. (b) Scanning electron microscopy image of the two devices B and D. (c) Differential conductance in logarithmic scale as a function of back-gate voltage for device A at zero applied bias. (d) Differential conductance of device A as a function of applied back-gate and bias voltage. (e) Closeup of a region of (d).

Figure 2

(a) Differential conductance as a function of applied back-gate and bias voltage for device B, (b) for device C, and (c) for device D.

Figure 3

(a) Current flowing through device A as a function of back-gate voltage and perpendicular magnetic field at a fixed bias of 0.1 mV. Cyan circles mark the approximate position at which the Coulomb-blockade peaks start to tilt. Black dashed lines mark the slope of the $\nu =2$ plateau for a micron-sized device and as a guide to the eye the predominant slope observed in this measurement. (b) Same plot as Fig. 1 added for direct comparison. (c), (d) Zooms into different regions of Fig. 2.

Figure 4

Schematic drawing of possible wave-function envelopes (red) that are localized in the constriction and couple to the extended wave functions in the graphene leads (blue). Charge carriers are primarily localized (a) in the full area of the constriction, (b) in the constriction and the right lead, (c) along the edge of the constriction, and (d) along the edge of the constriction and the right lead.

Figure 5

Results for the tight-binding simulation of a $30\times 30\mathrm{nm}$ graphene nanoconstriction connected to open leads of width $140\mathrm{nm}$, for different microscopic realizations of edge disorder. (a)–(f) Six exemplary eigenstates of the nanoconstrictions [see symbols in (g)]. (g) Statistics of the IPR [Eq. (1)] of 5000 eigenstates as a function of eigenenergy. States with large probability $P$ [Eq. (2)] to be inside the nanoconstriction are marked by blue triangles $\left(P>0.75\right)$ or red squares $(P\in [0.5,0.75]$, see the inset). Larger symbols denote the six states depicted in (a)–(f). In our simulation, the edge-localized states are slightly below the Dirac point (at $0\mathrm{eV})$ as a result of the third-nearest-neighbor coupling included in our tight-binding parametrization [58]. In the experiment, the exact on-site energies will also strongly depend on details of the local chemical environment at the graphene edge.