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Manipulation of edge states in microwave artificial graphene

Bellec, Matthieu, Kuhl, Ulrich, Montambaux, Gilles, Mortessagne, Fabrice·2014·New Journal of Physics
DOI 10.1088/1367-2630/16/11/113023
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Figure 1 (a) Picture of an unstrained artificial graphene ribbon with zigzag, bearded and armchair edges. The lattice constant is 15 mm. (b) Corresponding experimental density of states (DOS). The arrow indicates the zero-modes appearing at the Dirac frequency ν D .
Figure 1 (a) Picture of an unstrained artificial graphene ribbon with zigzag, bearded and armchair edges. The lattice constant is 15 mm. (b) Corresponding experimental density of states (DOS). The arrow indicates the zero-modes appearing at the Dirac frequency ν D .CC-BY · From the PDF
Figure 2 (top row). Experimental zero-mode intensities for β ranging from 0.4 to 2.5. The anisotropy axis is horizontal. Bearded and zigzag edges are displayed on the same lattice (respectively at the left and the right). (bottom row). Normalized intensities integrated over vertical lines. Green (resp. red) areas indicate qualitatively the bearded (resp. zigzag) zero-mode extension.
Figure 2 (top row). Experimental zero-mode intensities for β ranging from 0.4 to 2.5. The anisotropy axis is horizontal. Bearded and zigzag edges are displayed on the same lattice (respectively at the left and the right). (bottom row). Normalized intensities integrated over vertical lines. Green (resp. red) areas indicate qualitatively the bearded (resp. zigzag) zero-mode extension.CC-BY · From the PDF
Figure 3 Schematic representation of a semi-infinite lattices with zigzag ((a), red) and bearded ((b), green) edge along the ‖ axis. The anisotropy axis is horizontal, here along the ⊥ axis. The corresponding coupling strength is denoted t ′ . A n and B n are the wavefunction amplitudes in the two sublattices. k is the 1D Bloch wavevector (see text for details). (c) Zigzag - bearded edge states existence diagram for various k and β . The colormap corresponds to the inverse localization length in the zigzag (red, 1 /ξ zz ) and bearded (green, 1 /ξ bd ) cases; the darker the color, the stronger the states are localized. The dashed lines correspond to the values of β used in the experiments.
Figure 3 Schematic representation of a semi-infinite lattices with zigzag ((a), red) and bearded ((b), green) edge along the ‖ axis. The anisotropy axis is horizontal, here along the ⊥ axis. The corresponding coupling strength is denoted t ′ . A n and B n are the wavefunction amplitudes in the two sublattices. k is the 1D Bloch wavevector (see text for details). (c) Zigzag - bearded edge states existence diagram for various k and β . The colormap corresponds to the inverse localization length in the zigzag (red, 1 /ξ zz ) and bearded (green, 1 /ξ bd ) cases; the darker the color, the stronger the states are localized. The dashed lines correspond to the values of β used in the experiments.CC-BY · From the PDF
Figure 4 Stream and density plots of the phase φ ( k ‖ , k ⊥ ) in reciprocal space, for the three edges (zigzag, bearded and armchai) and for various deformations (see the sketches in figures 3 and 7). The first 2D (bulk) BZ is indicated by a white contour. The red dots are the positions of the Dirac points. The first 1D BZ associated to the ( ‖ ) and ( ⊥ ) directions are shown by solid and dashed black lines, respectively. For a given k ‖ , the existence of edge states is related to the winding of the phase along the k ⊥ direction in the BZ: there are edge states when the total rotation of the phase is 2 π as seen in the shaded areas [ Z ( k ‖ ) = π ]. There are no edge states otherwise [ Z ( k ‖ ) = 0].
Figure 4 Stream and density plots of the phase φ ( k ‖ , k ⊥ ) in reciprocal space, for the three edges (zigzag, bearded and armchai) and for various deformations (see the sketches in figures 3 and 7). The first 2D (bulk) BZ is indicated by a white contour. The red dots are the positions of the Dirac points. The first 1D BZ associated to the ( ‖ ) and ( ⊥ ) directions are shown by solid and dashed black lines, respectively. For a given k ‖ , the existence of edge states is related to the winding of the phase along the k ⊥ direction in the BZ: there are edge states when the total rotation of the phase is 2 π as seen in the shaded areas [ Z ( k ‖ ) = π ]. There are no edge states otherwise [ Z ( k ‖ ) = 0].CC-BY · From the PDF
Figure 5 Stream and density plots of the phase φ ( k ‖ , k ⊥ ) in reciprocal space, for the armchair edge and a deformation parallel to the edge. Marker and line legends are the same as in figure 4. The Zak phase is 0 everywhere, confirming the absence of edge state in this case.
Figure 5 Stream and density plots of the phase φ ( k ‖ , k ⊥ ) in reciprocal space, for the armchair edge and a deformation parallel to the edge. Marker and line legends are the same as in figure 4. The Zak phase is 0 everywhere, confirming the absence of edge state in this case.CC-BY · From the PDF
Figure 6 Experimental armchair zero-mode intensities for β ranging from 0.4 to 3.5. (a) β < 1. (b) β = 1, no edge states. (c - f) β > 1. Insets in (a) and (f) emphasize the switch between the two triangular sublattice excitation.
Figure 6 Experimental armchair zero-mode intensities for β ranging from 0.4 to 3.5. (a) β < 1. (b) β = 1, no edge states. (c - f) β > 1. Insets in (a) and (f) emphasize the switch between the two triangular sublattice excitation.CC-BY · From the PDF
Figure 7 (a) Schematic representation of a semi-infinite lattice with armchair edge along the ‖ axis. The anisotropy axis is horizontal, the corresponding coupling strength is denoted t ′ . A n and B n are the wavefunction amplitudes in the two sublattices. k is the Bloch wavevector (see text for details). (b) Armchair edge states existence diagram for various k and β . The colormap corresponds to the inverse localization length 1 /ξ ac . Light (dark) colors indicate a delocalized (localized) state. The white area indicates no edge states. For β > 1, only B -sites are excited. For β < 1, only A -sites are excited.
Figure 7 (a) Schematic representation of a semi-infinite lattice with armchair edge along the ‖ axis. The anisotropy axis is horizontal, the corresponding coupling strength is denoted t ′ . A n and B n are the wavefunction amplitudes in the two sublattices. k is the Bloch wavevector (see text for details). (b) Armchair edge states existence diagram for various k and β . The colormap corresponds to the inverse localization length 1 /ξ ac . Light (dark) colors indicate a delocalized (localized) state. The white area indicates no edge states. For β > 1, only B -sites are excited. For β < 1, only A -sites are excited.CC-BY · From the PDF
Figure 8 Experimental (a) and theoretical (b) armchair zero-mode intensity profiles for β ranging from 0.4 to 3.5. For β < 1 ( β > 1), A ( B ) sites are considered.
Figure 8 Experimental (a) and theoretical (b) armchair zero-mode intensity profiles for β ranging from 0.4 to 3.5. For β < 1 ( β > 1), A ( B ) sites are considered.CC-BY · From the PDF
Figure 9 (a) Armchair zero-mode amplitude components for β < 1 and k = 0. The wavefunction is non-zero only on A -sites (empty circles). m indicates the A -row index. The anisotropy axis is vertical, thick lines represent the corresponding coupling. (b) Zero-mode amplitude components for β > 2 and k = 0 appearing at the intersection of two zigzag edges.
Figure 9 (a) Armchair zero-mode amplitude components for β < 1 and k = 0. The wavefunction is non-zero only on A -sites (empty circles). m indicates the A -row index. The anisotropy axis is vertical, thick lines represent the corresponding coupling. (b) Zero-mode amplitude components for β > 2 and k = 0 appearing at the intersection of two zigzag edges.CC-BY · From the PDF

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Cites

Show all 9 references

Research works printed but not linked: 26

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Metadata

Authors
Bellec, M., Kuhl, U., Montambaux, G., & Mortessagne, F.
Venue
New Journal of Physics
Publisher
IOP Publishing
Published
2014-11-11
Volume
16
Issue
11
Pages
113023
ISSN
1367-2630
DOI
10.1088/1367-2630/16/11/113023
Cited by
65
References
9
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