Graphene Quantum Dots and Wavefunction Imaging

The Crommie group has developed a new technique for locally gating nanoscale regions of atomically-clean graphene by injecting charge directly into the boron nitride (BN) insulator just below the graphene.1,2 Both polarities of charge can be independently implanted at different locations in the insulator by using a scanning tunneling microscope (STM) tip. This allows nanoscale confinement potentials to be “drawn” directly into graphene, thus allowing the graphene electronic wavefunction to be directly manipulated. The resulting graphene wavefunctions can then be imaged using STM spectroscopic techniques.2

Fig. 1: (a) Process of injecting negative charge into BN insulator below graphene to form local gate. (b) Sketch of resulting local p-doped area in graphene right above BN region where negative charge was injected.

Fig. 1a shows a sketch of the process of injecting negative charge into a BN insulator under graphene, which then causes the graphene overlayer to have a local p-doped region as shown in Fig. 1b. The results of the experimental process are shown in Fig. 2. Here the graphene is seen before charge injection in Fig. 2a and after charge injection in Fig. 2b (charge was injected at the right upper hand corner). This process creates a circular p-n junction as sketched in the inset of Fig. 2b.

Fig. 2: (a) STM image of pristine graphene above BN before charge injection to form local gate. (b) STM image of same region after injecting charge into BN at upper right corner. This creates a circular p-n junction in the graphene. (STM data from Crommie group).

Fig. 3 shows an image of the quantum mechanical eigenstate that is experimentally measured both inside and outside of the graphene quantum dot. A circular nodal pattern is observed inside the dot, as well as Friedel-like oscillations in the exterior region.

Fig. 3: STM image of Dirac fermion wavefunction both inside and outside of quantum dot formed by local gate injection. (STM data from Crommie group).

A larger-scale view of the dot can be seen in Fig. 4. This can be thought of as a relativistic quantum corral,3 since the confined electrons are now Dirac fermions. The energy-dependent nodal structure of the dot is shown in Fig. 5a.

Fig 4: Larger scale STM image ofDirac fermion wavefunction in circular quantum dot formed in graphene using local gate injection technique. (STM data from Crommie group).

We are able to model the quantum mechanical behavior of these quantum dots by solving the Dirac equation in the presence of a 2D parabolic confinement potential (theory performed by L. Levitov’s group). This is essentially a 2D relativistic simple harmonic oscillator. The fit is good (Fig. 2b), and shows that the nodal structure can be explained by a hierarchy of principal and total angular momentum quantum numbers.2 Here the total angular momentum is the sum of orbital angular momentum and pseudospin.

Fig 5: (a) Spectroscopic map shows energy nodal structure of Dirac fermion eigenstates measured inside graphene quantum dot using STM spectroscopy. R=0 is the center of the quantum dot. (b) Theoretical prediction of quantum dot energy nodal structure obtained by solving the Dirac equation for a parabolic well. (STM data: Crommie group; theory: Levitov group).

This new quantum dot fabrication technique is quite general and the Crommie group is exploring its application to other 2D materials.

References

1. J. Velasco, L. Ju, D. Wong, S. Kahn, J. Lee, H.-Z. Tsai, C. Germany, S. Wickenburg, J. Lu, T. Taniguchi, K. Watanabe, A. Zettl, F. Wang & M. F. Crommie, "Nanoscale Control of Rewriteable Doping Patterns in Pristine Graphene/Boron Nitride Heterostructures", Nano Letters 16, 1620 (2016).

2. J. Lee, D. Wong, J. Velasco Jr, J. F. Rodriguez-Nieva, S. Kahn, H.-Z. Tsai, T. Taniguchi, K. Watanabe, A. Zettl, F. Wang, L. S. Levitov & M. F. Crommie, "Imaging electrostatically confined Dirac fermions in graphene quantum dots", Nature Physics 12, 1032 (2016).

3. M. F. Crommie, C. P. Lutz & D. M. Eigler, "Confinement of Electrons to Quantum Corrals on a Metal Surface", Science 262, 218 (1993).