Frustrated Collapse in Coulomb Arrays
Atomic collapse is an intrinsically relativistic phenomenon that occurs when enough charge (i.e., Z > Zc) is concentrated at one location to make the Coulomb potential strong enough to cause electrons to spiral down to the center of the potential well1,2,3 (sort of like an electronic black hole). (For a description of the Crommie group’s observation of atomic collapse for single-impurity Coulomb potentials please look here. The Crommie group has also explored the less well-understood (but more physically common) case where charge centers are distributed over a wide area in space.4 This situation is surprisingly similar to the gravitational lensing of light seen in astronomical observations of large, cosmological mass distributions.5 The electrons of graphene play the role of light here since they have a linear dispersion relation similar to light (except they are slower: vF = c/300). The Coulomb potential of a charged impurity in graphene plays the role of the gravity well of a star -- charged impurities scattered throughout graphene are thus analogous to stars scattered throughout space. The semiclassical motion of electrons near a cluster of charged impurities in graphene is formally analogous to the motion of photons in the vicinity of a star cluster. If the charged impurities in graphene are each subcritical (Z < Zc) then this is equivalent to the case where none of the stars are black holes. On the other hand, if the total charge of all the charge centers exceeds the critical charge (Σi Zi > Zc) then this is analogous to the case where the extent of the mass distribution is close to the Schwarzchild radius.5 In this case an electron will follow a semiclassical trajectory where, in the far field, it collapses to the center of the charge distribution (Fig. 1a). Once “inside” the impurity cluster, however, the fall to the center is frustrated by the presence of subcritical impurities and the electron is scattered and meanders between impurities (Fig. 1b), similar to a trapped light beam (theory by V. Pereira and A. Tatan).
In order to observe this phenomenon experimentally, the Crommie group prepared atomically-precise 1D arrays of charged Coulomb centers on a graphene FET device (Fig. 2). This was performed by first depositing PCDA molecules onto the graphene to create molecular anchor points (PCDA is inert and acts as a nanoscale dielectric) (Figs. 2b, c) ) whose density can be controllably varied.4 F4TCNQ was chosen for this study because its charge state can easily be switched between Q = 0 and Q = -e by electrostatically gating the graphene FET.6 The charge and distribution of Coulomb impurities can thus be controlled with unprecedented precision.
We then performed STM spectroscopy near the charged 1D arrays. For dilute arrays where the charged molecules are far apart, no spectroscopic features were seen near the molecules (Figs. 3a, b). This is consistent with the fact that each charged molecule is a subcritical impurity (Z < Zc). For dense arrays, however, an extended quasi-bound state formed at an energy near the graphene Dirac point (Figs. 3c-e), similar to the atomic collapse states that we observed previously around supercritical impurities.1 An important difference, though, is that the charges here are supercritical only in a collective sense (since each individual charge is subcritical and they are not clustered together into a single “artificial nucleus” as was done in the previous study). This new behavior is the realization of frustrated supercriticality because an electron far from the array collapses toward it and “sees” it as a
supercritical impurity, but as the electron nears the array its fall to the center is “frustrated” by the fact that the individual charges in the array are subcritical, and so the electron is eventually scattered by the individual impurities in the near field (Fig. 1)4 We performed a quantum mechanical simulation of this behavior using tight-binding theory and were able to calculate frustrated collapse eigenstates that match our data well (Fig. 4). It is striking that we are able to gain insight into nanoscale electronic phenomena by comparing it to astronomical phenomena occurring at cosmological length scales.
1. Y. Wang, D. Wong, A. V. Shytov, V. W. Brar, S. Choi, Q. Wu, H.-Z. Tsai, W. Regan, A. Zettl, R. K. Kawakami, S. G. Louie, L. S. Levitov & M. F. Crommie, "Observing Atomic Collapse Resonances in Artificial Nuclei on Graphene", Science 340, 734 (2013).
2. A. V. Shytov, M. I. Katsnelson & L. S. Levitov, "Atomic Collapse and Quasi--Rydberg States in Graphene", Physical Review Letters 99, 246802 (2007).
3. V. M. Pereira, J. Nilsson & A. H. Castro Neto, "Coulomb Impurity Problem in Graphene", Physical Review Letters 99, 166802 (2007).
4. J. Lu, H.-Z. Tsai, A. N. Tatan, S. Wickenburg, A. A. Omrani, D. Wong, A. Riss, E. Piatti, K. Watanabe, T. Taniguchi, A. Zettl, V. M. Pereira & M. F. Crommie, "Frustrated supercritical collapse in tunable charge arrays on graphene", Nature Communications 10, 477 (2019).
5. Supplementary Information of ref. 4.
6. S. Wickenburg, J. Lu, J. Lischner, H.-Z. Tsai, A. A. Omrani, A. Riss, C. Karrasch, A. Bradley, H. S. Jung, R. Khajeh, D. Wong, K. Watanabe, T. Taniguchi, A. Zettl, A. H. C. Neto, S. G. Louie & M. F. Crommie, "Tuning charge and correlation effects for a single molecule on a graphene device", Nature Communications 7, 13553 (2016).