1. Electronic states of finite length carbon nanotubes Yuki Tatsumi, Wataru Izumida Tohoku University, Department of Physics Outline  Background “SWNT quantum dot” ・ Four-fold degeneracy & two-fold degeneracy ・ Vernier spectrum  Motivation  Electronic states of armchair SWNTs → “1D ladder model”  Result “vernier spectrum”  Summary 200nm nanotube S. Sapmaz et al.; nature429, p389-392 (2004) Electrode(source) Electrode(drain)

2. Carbon nanotube as a quantum dot Schematic of a quantum dot Nanotube quantum dot S. Sapmaz et al.; Phys. Rev. B 71, 153402 (2005) Coulomb oscillation Valley degeneracy(K, K’) Spin degeneracy(↑, ↓) Fourfold degeneracy 200nm nanotube S. Sapmaz et al.; nature429, p389-392 (2004) fourfold =0 (If degenerate) Electrode (source) Electrode (drain)

3. Two-fold & four-fold degeneracy A. Makarovski et al,: Phys. Rev. B 74, 155431 (2006) Twofold degeneracy? Two-fold? Four-fold? Fourfold degeneracy Twofold degeneracy? ・・・ Periodically? A. Makarovski et al,: Phys. Rev. B 74, 155431 (2006) =0 (If degenerate) Fourfold BUT

4. “Vernier” spectrum ? W. Izumida et al,: Phys. Rev. B 85, 165430 (2012) “Vernier” spectrum Energy level of QD 2- or 4-fold degeneracy Right-going@K Left-going@K’ Energy band tilting SWNT curvature What is the electronic states in finite length carbon nanotubes? Motivation Standing wave ・・・ K-left-going + K’-right-going ? ππ Quantum dot

5. 1D ladder model for armchair SWNTs L. Balents, et al,; Phys. Rev. B, 55, R11973 (1996) Armchair tube Nearest neighbor Second nearest neighbor Nearest neighbor Second neighbor Method ・ Open boundary condition ・ Tight binding method Calculate this model !! 1D Ladder model Tilting effect KK’ Second nearest also …

6. (eV) eigenenergy Right-going Left-going Vernier spectrum Result : vernier spectrum for 1D ladder model Armchair SWNT, diametr 0.8nm, length 200nm fourfold twofold fourfold 2 and 4 fold degeneracy

7. Summary  Vernier spectrum of 1D Ladder model (armchair nanotube model) → Two and Four fold degeneracy A. Makarovski et al,: Phys. Rev. B 74, 155431 (2006) twofold fourfold

8. Armchair → Ladder model Nearest neighborSecond neighbor

9. fourfold twofold