Limits...
Flat edge modes of graphene and of Z2 topological insulator.

Imura K, Mao S, Yamakage A, Kuramoto Y - Nanoscale Res Lett (2011)

Bottom Line: We argue that the appearance of such edge modes are naturally understood by regarding graphene as the gapless limit of a Z2 topological insulator.To illustrate this idea, we consider both Kane-Mele (graphene-based) and Bernevig-Hughes-Zhang models: the latter is proposed for HgTe/CdTe 2D quantum well.Much focus is on the role of valley degrees of freedom, especially, on how they are projected onto and determine the 1D edge spectrum in different edge geometries.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Quantum Matter, AdSM, Hiroshima University, Higashi-Hiroshima, 739-8530, Japan. imura@hiroshima-u.ac.jp.

ABSTRACT
A graphene nano-ribbon in the zigzag edge geometry exhibits a specific type of gapless edge modes with a partly flat band dispersion. We argue that the appearance of such edge modes are naturally understood by regarding graphene as the gapless limit of a Z2 topological insulator. To illustrate this idea, we consider both Kane-Mele (graphene-based) and Bernevig-Hughes-Zhang models: the latter is proposed for HgTe/CdTe 2D quantum well. Much focus is on the role of valley degrees of freedom, especially, on how they are projected onto and determine the 1D edge spectrum in different edge geometries.

No MeSH data available.


Δ/B = 3.2.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3211448&req=5

Figure 8: Δ/B = 3.2.

Mentions: The edge spectrum in the straight edge geometry is obtained analytically as [21,23], E(kx) = ± A sin kx. As is clear from the expression, the spectrum does not depend on Δ/B, which is very peculiar to the straight edge case. Only the range of the existence of edge modes changes as a function of Δ/B (see Figures 5, 6, 7, 8, 9 and 10) [21]. In the figure, the energy spectrum (of edge + bulk modes) obtained numerically for a system of 100 rows is shown in a ribbon geometry with two straight edges. Starting with Figure 5 (spectrum shown in red), the value of Δ/B is varied as Δ/B = 0.2, Δ/B = 0.8 (green, Figure 6), Δ/B = 2 (blue, Figure 7), Δ/B = 3.2 (cyan, Figure 8), and Δ/B = 4 (magenta, Figure 9). All of these five plots are superposed in the last panel (Figure 10). A and B are fixed at unity. The dotted curve is a reference showing the exact edge spectrum. The plots show explicitly that the edge spectrum at different values of Δ/B are indeed on the same sinusoidal curve.


Flat edge modes of graphene and of Z2 topological insulator.

Imura K, Mao S, Yamakage A, Kuramoto Y - Nanoscale Res Lett (2011)

Δ/B = 3.2.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3211448&req=5

Figure 8: Δ/B = 3.2.
Mentions: The edge spectrum in the straight edge geometry is obtained analytically as [21,23], E(kx) = ± A sin kx. As is clear from the expression, the spectrum does not depend on Δ/B, which is very peculiar to the straight edge case. Only the range of the existence of edge modes changes as a function of Δ/B (see Figures 5, 6, 7, 8, 9 and 10) [21]. In the figure, the energy spectrum (of edge + bulk modes) obtained numerically for a system of 100 rows is shown in a ribbon geometry with two straight edges. Starting with Figure 5 (spectrum shown in red), the value of Δ/B is varied as Δ/B = 0.2, Δ/B = 0.8 (green, Figure 6), Δ/B = 2 (blue, Figure 7), Δ/B = 3.2 (cyan, Figure 8), and Δ/B = 4 (magenta, Figure 9). All of these five plots are superposed in the last panel (Figure 10). A and B are fixed at unity. The dotted curve is a reference showing the exact edge spectrum. The plots show explicitly that the edge spectrum at different values of Δ/B are indeed on the same sinusoidal curve.

Bottom Line: We argue that the appearance of such edge modes are naturally understood by regarding graphene as the gapless limit of a Z2 topological insulator.To illustrate this idea, we consider both Kane-Mele (graphene-based) and Bernevig-Hughes-Zhang models: the latter is proposed for HgTe/CdTe 2D quantum well.Much focus is on the role of valley degrees of freedom, especially, on how they are projected onto and determine the 1D edge spectrum in different edge geometries.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Quantum Matter, AdSM, Hiroshima University, Higashi-Hiroshima, 739-8530, Japan. imura@hiroshima-u.ac.jp.

ABSTRACT
A graphene nano-ribbon in the zigzag edge geometry exhibits a specific type of gapless edge modes with a partly flat band dispersion. We argue that the appearance of such edge modes are naturally understood by regarding graphene as the gapless limit of a Z2 topological insulator. To illustrate this idea, we consider both Kane-Mele (graphene-based) and Bernevig-Hughes-Zhang models: the latter is proposed for HgTe/CdTe 2D quantum well. Much focus is on the role of valley degrees of freedom, especially, on how they are projected onto and determine the 1D edge spectrum in different edge geometries.

No MeSH data available.