Electronic states of finite length carbon nanotubes
Yuki Tatsumi, Wataru IzumidaTohoku 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)
Carbon nanotube as a quantum dotSchematic of a quantum dot
Nanotube quantum dot
S. Sapmaz et al.; Phys. Rev. B 71, 153402 (2005)Addition energy
Coulomb oscillation
Valley degeneracy(K, K’)Spin degeneracy(↑, ↓)
Fourfold degeneracy
200nm
nanotube
S. Sapmaz et al.; nature429, p389-392 (2004)
fourfold
Peak distance Addition energy
=0 (If degenerate)
: Chemical potential : N-th energy level : Coulomb energy with other electron
Electrode(source)
Electrode(drain)
Two-fold & four-fold degeneracy
Twofold degeneracy?
Two-fold? Four-fold?
Fourfold degeneracy Twofold degeneracy? ・・・ Periodically?
A. Makarovski et al,: Phys. Rev. B 74, 155431 (2006)
=0 (If degenerate)
Peak distance
Fourfold
BUT
“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 tiltingSWNT curvature
What is the electronic states in finite length carbon nanotubes?
Motivation
Standing wave ・・・ K-left-going + K’-right-going
?
π π
Quantum dot
𝐻nn=∑𝒌
¿¿
𝐶𝐴 𝒌
+¿= 1√𝑁 𝑦
∑𝑦𝑒𝑖 𝑘𝑦 𝑦𝐶𝐴𝑘
𝑥𝑦+¿¿ ¿
𝐶𝐵𝒌=1
√𝑁 𝑦
∑𝑦
𝑒−𝑖𝑘 𝑦 𝑦𝐶𝐵𝑘𝑥 𝑦
Partially ( only) Fourier transformation
1D ladder model for armchair SWNTsL. Balents, et al,; Phys. Rev. B, 55, R11973 (1996)
Only Cutting line
Armchair tube
Nearest neighbor
Second nearest neighbor
𝑘𝑥=0
Nearest neighbor Second neighbor
Method・ Open boundary condition・ Tight binding method
Calculate this model !!
1D Ladder model
Tilting effect
K K’
Second nearest also …
(eV)
eigenenergy
Right-going Left-going
Vernier spectrum
Result : vernier spectrum for 1D ladder model
Armchair SWNT, diametr 0.8nm, length 200nm
𝐸addition=(𝜀𝑁+1−𝜀𝑁 )+𝑈 coulomb
←
fourfold
fourfold
twofold
twofold
twofold twofoldfourfold
2 and 4 fold degeneracy
SummaryVernier spectrum of 1D Ladder model (armchair
nanotube model)
→ Two and Four fold degeneracy
A. Makarovski et al,: Phys. Rev. B 74, 155431 (2006)
𝐸addition=(𝜀𝑁+1−𝜀𝑁)+𝑈 coulomb
twofold twofold
twofold twofold
fourfold
fourfold
Armchair → Ladder model𝑯=𝑯𝟏+𝑯𝟐
𝑯𝟐= ∑𝒏 , 𝒊=𝑨 ,𝑩
¿¿
Nearest neighbor Second neighbor
fourfold
fourfold
twofold
twofold