Coulomb versus spin-orbit interactionin carbon-nanotube quantum dots

Andrea Secchi and Massimo RontaniCNR-INFM Research Center S3 and University of Modena,

Modena, Italy

• exact diagonalization of few-electron Hamiltonian

• clarification of recent tunneling experiments

Carbon-nanotube quantum dots

quasi-1D systems

double degeneracyF. Kuemmeth et al., Nature 452, 448 (2008)

Strong correlation or not in CN QDs?

Strong correlation or not in CN QDs?

spin-orbit interaction splits 4-fold degenerate spin-orbitals

Low temperature SETS experiment

spin isospin

Strong correlation or not in CN QDs?

two-electron ground state:

one Slater determinant

no correlation chem

ical

pote

nti

al

the simplest interpretation

CI model: 1D harmonic potential

theory exp configuration-interaction (CI) calculation:

two valleys

QD: harmonic potential

forward & backward Coulomb interactions

spin-orbit coupling

free parameter:

M. Rontani et al., JCP 124, 124102 (2006)

Strongly correlated CI wave functions

A & B states:

strongly correlated

same orbital wave functions

differ in isospin only

A. Secchi and M. Rontani, arXiv: 0903.5107

isospin = valley population

different harmonic oscillator quantum numbers

Independent-particle feature explained

exp

A. Secchi and M. Rontani, arXiv: 0903.5107

N = 2

N = 1

B(T)

theo

A and B:

correlated

T3 = 0, 1

split by spin-orbit int. only

T3 = 0 T3 = 1

T3 = 1/2

T3 = -1/2

Non-universal tunneling spectrum

exp meV80 meV40

A. Secchi and M. Rontani, arXiv: 0903.5107

N = 2

N = 1

CI two-electron energy spectrum

A. Secchi and M. Rontani, arXiv: 0903.5107

0

gerade

ungerade

x

n(x)

Pair correlation functions

g(X) = probability to find a couple of electrons at relative distance X

0

dXXgXgd ACCA

Conclusions

• spin-orbit and Coulomb interactions coexist

• non-interacting features of tunneling spectra explained • we predict electrons to form a Wigner molecule

andrea.secchi@unimore.itmassimo.rontani@unimore.it

www.s3.infm.itwww.nanoscience.unimore.it/max.html

Single-particle Hamiltonian

Bloch states in K and K’ valleys

envelope function

spin-orbit interaction and magnetic field

Effective 1D Coulomb interaction

Ohno potential

trace out x and z degrees of freedom

forward

backward

Fully interacting Hamiltonian

Spin-orbit coupling for two electrons

six-fold degenerate

zzSO RH ˆˆ)/(ˆSO

Wigner-Mattis theorem is not appliable in nanotubes

),(),(),;,( 21212211 zzzz xxxx

),( 21 xx nodeless in the ground state

),(),( 1221 xxxx

),(),( 1221 zzzz S = 0

isospin T = additional degree of freedom

),(),(),(),,;,,( 212121222111 zzzzzzzz xxxx

either (S = 0, T = 1) or (S = 1, T = 0)Tz = -1, 0, +1 Sz = -1, 0, +1