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