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. 2017 Jun 22:8:15961.
doi: 10.1038/ncomms15961.

Imaging quasiperiodic electronic states in a synthetic Penrose tiling

Laura C Collins  1 , Thomas G Witte  1 , Rochelle Silverman  1 , David B Green  1 , Kenjiro K Gomes  1
Affiliations

Imaging quasiperiodic electronic states in a synthetic Penrose tiling

Laura C Collins et al. Nat Commun. .

Abstract

Quasicrystals possess long-range order but lack the translational symmetry of crystalline solids. In solid state physics, periodicity is one of the fundamental properties that prescribes the electronic band structure in crystals. In the absence of periodicity and the presence of quasicrystalline order, the ways that electronic states change remain a mystery. Scanning tunnelling microscopy and atomic manipulation can be used to assemble a two-dimensional quasicrystalline structure mapped upon the Penrose tiling. Here, carbon monoxide molecules are arranged on the surface of Cu(111) one at a time to form the potential landscape that mimics the ionic potential of atoms in natural materials by constraining the electrons in the two-dimensional surface state of Cu(111). The real-space images reveal the presence of the quasiperiodic order in the electronic wave functions and the Fourier analysis of our results links the energy of the resonant states to the local vertex structure of the quasicrystal.

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Conflict of interest statement

The authors declare no competing financial interests.

Figures

Figure 1
Figure 1. Synthetic Penrose tiling quasicrystal.
(a) STM topograph of assembled quasicrystal composed of 460 CO molecules measured at a bias voltage V=10 mV and setpoint current I=1 nA. The CO molecules are located at the centre of each dark spot in the topograph. The overlay on the right side is the Penrose tiling composed of rhombi with side length a0=1.6 nm and vertices angles 72°/108° (blue) and 36°/144° (green). Scale bar, 4 nm. (b) Atlas of the eight types of vertex sites encountered in the Penrose vertex model tiling.
Figure 2
Figure 2. Visualizing a quasicrystalline electronic state.
(a) Topograph of the assembled quasicrystal composed of 460 CO molecules (dark spots in the image) measured with a bias voltage V=500 mV and setpoint current I=500 pA over a 42 nm by 42 nm field of view. Scale bar, 5 nm. (b) Normalized differential conductance map of the same field of view as a measured at bias voltage V=0 mV. (c) Histogram of the normalized differential conductance at bias voltage V=0 mV averaged over each type of vertex site. (d) Diagram of all vertex sites of the quasicrystal displaying the same field of view as a with site-B locations highlighted in cyan. (e) Fourier transform of the conductance map displayed in b with the arrow representing a wave-number value equal to twice the Fermi wave-number of the bare copper surface states (kF). Scale bar, 4 nm−1. (f) Fourier transform of quasicrystal model structure built with the same proportions as the one used in the experiment but with tens of thousands of sites to improve sharpness of Fourier peaks. The arrow represents the same wavenumber (2kF) as in e.
Figure 3
Figure 3. Topologically distinct sites.
(a) Normalized differential conductance map of a 42 nm by 42 nm field of view measured at bias voltage V=−200 mV. Scale bar, 5 nm. (b) Conductance map of the same field of view but at bias voltage V=−100 mV. (c) Diagram of all vertex sites of the quasicrystal displaying the same field of view as above and with site-E, site-G and site-H highlighted in cyan. (d) Diagram of all vertex sites of the quasicrystal displaying the same field of view as above and with site-F highlighted in cyan. (e) Histogram of the normalized differential conductance at bias voltage V=−200 mV (left) and V=−100 mV (right) averaged over each type of vertex site.
Figure 4
Figure 4. Tunnelling spectroscopy survey.
(a) The grey diagram illustrates the Penrose tiling assembled in our experiment. The colour dots mark the 61 electron sites where we measured the differential conductance spectra, with each colour corresponding to a site type. (b) The red line is the average of all 61 differential conductance spectra measure in the quasicrystal and the grey line a differential conductance spectra of the bare Cu (111) surface, spatially averaged over 100 distinct points. (c) The total normalized conductance spectra, calculated by the ratio of the two spectra presented above. (d) Normalized conductance spectra averaged by site type for all eight sites in the Penrose tiling. The y-scale refers to the bottom spectra taken at the A-sites. Other spectra have been offset for clarity by 0.3 units in the y-scale above the previous spectrum.
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