shortened version

Modeling Graphene Layers and Single-walled Carbon Nanotubes with Regularized δ-function Potentials : 

Ilya Prigogine Center for Studies in Statistical Mechanics & Complex Systems http://order.ph.utexas.edu Modeling Graphene Layers and Single-walled Carbon Nanotubes with Regularized δ-function Potentials Han Hsu and L. E. Reichl Center for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, Texas 78712 Published on Phys. Rev. B 72, 155413 (2005)

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Outlines Motivation Divergence and regularization of two-dimensional δ-function potential δ-potential model for graphene layer and single-walled nanotubes (SWNT) Summary

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Motivation To study SWNT under intense laser fields. (Band dynamics, high harmonic generation, ...) Perturbation theory is no longer valid!! Carrying out ab initio calculation in nonperturbative regime is very difficult. Need a simple model for SWNT that still provides some accuracy.

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Graphene and SWNT A single-walled carbon nanotube (SWNT) can be imagined as a rolled-up rectangular strip of a graphene layer. (Cited from Charles M. Lieber group)

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Chiral Vectors and SWNT An SWNT with a chiral vector armchair: n = m zigzag: m = 0 chiral: otherwise http://www.seas.upenn.edu/mse/research/nanotubes.html is called an (n,m) nanotube.

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δ-potential Honeycomb Lattice Replace the atomic potential by a two-dimensional attractive δ-function Vatom(r) = Voδ(r), Vo < 0.

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Truncation of Basis Set

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Band Structures of Various Nt Increasing Nt makes the potential deeper.

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Band Structure of Graphene: ab initio, δ-potential, and TB Models The maximum difference in the optical energy range is 0.09 eV. δ-potential is closer to ab initio than nearest-neighbor TB (Saito & Dresselhaus). It compares with 3rd-neighbor TB. Ab initio & 3rd-neighbor TB: S. Reich et al., Phys. Rev. B 66, 035412 (2002)

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Band Structure of (10, 10) SWNT: δ-potential and 3rd-TB / ab initio δ-potential and 3rd-TB/ab initio are almost the same. 3rd-neighbor TB: S. Reich et al., Phys. Rev. B 66, 035412 (2002)

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Band Structure of (19, 0) SWNT: δ-potential and 3rd-TB / ab initio δ-potential and 3rd-TB/ab initio are almost the same. The slight discrepancy appears in high-energy states. 3rd-neighbor TB: S. Reich et al., Phys. Rev. B 66, 035412 (2002)

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Band Structure of (12, 3) SWNT: δ-potential and 3rd-TB / ab initio δ-potential and 3rd-TB/ab initio are almost the same. The slight discrepancy appears in high-energy states. 3rd-neighbor TB: S. Reich et al., Phys. Rev. B 66, 035412 (2002)

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Cosine-like Standing Wave HOMO LUMO Ab initio calculation for (6, 6) SWNT, L = 4.18nm Angel Rubio et el., Phys. Rev. Lett. 82, 3520 (1999) δ-potential model: (10, 10) SWNT with L=25a Close-end standing wave Phys. Rev. B 72, 155413 (2005)

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Sine-like Standing Wave HOMO LUMO Ab initio calculation for (6, 6) SWNT, L = 4.06nm Angel Rubio et el., Phys. Rev. Lett. 82, 3520 (1999) δ-potential model: (10, 10) SWNT with L=25a Open-end Standing wave Phys. Rev. B 72, 155413 (2005)

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Summary Regularized δ-potential is simple. The band structure and wave functions provided by δ-potential model agree with ab initio calculations. δ-potential model can be a good candidate to simulate the electrons of SWNTs under intense laser fields. More information in Phys. Rev. B 72, 155413 (2005)