Quantum-Confined Electronic States in Atomically Well-Defined Graphene Nanostructures

Despite the enormous interest in the properties of graphene and the potential of graphene nanostructures in electronic applications, the study of quantum-confined states in atomically well-defined graphene nanostructures remains an experimental challenge. Here, we study graphene quantum dots (GQDs) with well-defined edges in the zigzag direction, grown by chemical vapor deposition on an Ir(111) substrate by low-temperature scanning tunneling microscopy and spectroscopy. We measure the atomic structure and local density of states of individual GQDs as a function of their size and shape in the range from a couple of nanometers up to ca. 20 nm. The results can be quantitatively modeled by a relativistic wave equation and atomistic tight-binding calculations. The observed states are analogous to the solutions of the textbook “particle-in-a-box” problem applied to relativistic massless fermions.

Figure 1

STM imaging and spectroscopy on GQDs grown by CVD on Ir(111). (a) Large-scale STM image of graphene islands (G) on an Ir(111) substrate (acquired at $I=40\mathrm{pA}$ and $V_b=1.0\mathrm{V}$). Small graphene QDs have been indicated by red circles. (b) STM topography of a small GQD ($-0.05\mathrm{V}$ and 100 pA) with an overlaid atomic model which has perfect hexagonal symmetry with 7 benzene ring long edges. (c) $d\ln I/d\ln V_b$ spectra measured on the points indicated in (b), the green bars indicate the bias voltages corresponding to the LDOS maps shown in (d). (d) Measured LDOS maps (gray line denotes the edges of the GQD) at bias voltages corresponding to the two resonances in the spectra shown in (c). (e) The corresponding LDOS maps calculated for a particle in a box at the indicated energies and the underlying eigenstates.

Figure 2

Detailed comparison between STM and STS experiments and computational results on a large GQD. (a) Atomically resolved STM image of the GQD ($I=3\mathrm{nA}$, $V_b=1\mathrm{mV}$). (b) $dI/d{V}_b$ maps recorded under constant-current STM feedback at the bias voltages indicated in the figure ($I=1\mathrm{nA}$). (c),(d) Corresponding LDOS plots at the indicated energies calculated using a TB model (c) and the KG equation (d) as described in the text. (e) Correspondence between the experimental and the calculated energies based on TB (red squares) and the KG equation (blue circles) calculated with $v_F=6.2\times {10}^5\mathrm{m}/\mathrm{s}$.

Figure 3

Electronic structure of GQDs as a function of their size. (a) STM sample bias corresponding to the $S$ state as a function of the area $A$ of the GQD. The solid and dashed lines are fits to $1/A^{1/2}$ and $1/A$ scaling, respectively. (b) Composition of the STM topographies of the GQDs used in (a) (with different scan parameters). (c) A plot of the bias voltages from the STM experiments ($x$ axis) and the energies that give comparable LDOS calculated from the Klein-Gordon equation using a single value for $v_F=6.2\times {10}^5\mathrm{m}/\mathrm{s}$ ($y$ axis).