Wigner molecules in carbon-nanotube quantum dots

Wigner moleculesin carbon-nanotube quantum dots

Massimo Rontani and Andrea SecchiS3, Istituto di Nanoscienze – CNR, Modena, Italy

ultraclean semiconducting nanotubes

Bockrath group, Nature Phys. 2008 McEuen group, Nature 2008

gate-defined quantum dots

shallow confinement potentials (approx. parabolic)

ultraclean semiconducting nanotubesMcEuen group, Nature 2008

chem

ical p

ote

nti

al (

N)Bockrath group, Nature Phys. 2008

chem

ical p

ote

nti

al (

N)

0 8B (T)

1h

3h

5h

B (T)B (T)

1e

2e

3e

20 2

*)(

mRg

BEN B

ultraclean semiconducting nanotubesBockrath group, Nature Phys. 2008 McEuen group, Nature 2008

chem

ical p

ote

nti

al (

N)

chem

ical p

ote

nti

al (

N)

0 8B (T)

1h

3h

5h

B (T)B (T)

1e

2e

3e

20 2

*)(

mRg

BEN B

)1()( 000 NENEE

independent from B


chem

ical p

ote

nti

al (

N)

chem

ical p

ote

nti

al (

N)

0 8B (T)

1h

3h

5h

B (T)B (T)

1e

2e

3e

20 2

*)(

mRg

BEN B

)1()( NN spin added electron


chem

ical p

ote

nti

al (

N)

chem

ical p

ote

nti

al (

N)

0 8B (T)

1h

3h

5h

B (T)B (T)

1e

2e

3e

20 2

*)(

mRg

BEN B

)1()( NN isospin added el.(angular momentum)


chem

ical p

ote

nti

al (

N)

chem

ical p

ote

nti

al (

N)

0 8B (T)

1h

3h

5h

B (T)B (T)

1e

2e

3e

20 2

*)(

mRg

BEN B

ground statespin & isospinpolarized

Wigner molecule?

single-particle + spin-orbit

motivation

Coulomb interaction vs single-particle physics

role of interaction?

exps at Harvard and Delft on coherent spin manipulation

outlook (I)

similar issues for graphene quantum dots

similar theoretical approach (see next slide)

Hamiltonian

exact diagonalisation ground & excited states

many-body term: Ohno potential, inter- and intra-valley channels (including short range terms)many-body term: Ohno potential, inter- and intra-valley channels (including short range terms)

compute the wavefunction as a superposition of Slater compute the wavefunction as a superposition of Slater

determinantsdeterminants ij

i

Ni

NiN HHc 0|| †

''† ml

Ni cc

Rontani et al., J. Chem. Phys. 124, 124102 (2006)

single-particle term: mass + isospin + 1D harmonic confinement + single-particle term: mass + isospin + 1D harmonic confinement + BB + spin-orbit coupling + spin-orbit coupling

compute compute ((NN), ), nn((xx), ), gg((xx),… ),…

envelope function envelope function approximationapproximationLuttinger and Kohn 1955, Ando 2005

)(),,()();,,( szyxxFszyx nn

)2()1( ˆˆˆ VHH

experimental evidence

split 4-fold degenerate spin-orbitals

non-interacting physics?

two-electron ground state:

one Slater determinant

no correlation chem

ical

pote

nti

al

the simplest interpretation

theory vs experiment

theoryPRB 80, 041404(R) (2009) McEuen group 2008

B (T)

dielectric constant

fitting parameter

strongly correlated wave functions

A & B states:

strongly correlated

same orbital wave functions

differ in isospin only

A. Secchi and M.R., PRB 80, 041404(R) (2009)

isospin = valley population

spectrum affected by interaction

N = 2

N = 1

A. Secchi & M.R., PRB 80, 041404(R) (2009)

interaction strength

SO

SO

crystallization criterionA. Secchi & M.R., PRB 82, 035417 (2010) Bockrath group, Nature Phys. 2008

chem

ical p

ote

nti

al (

N)

0 8B (T)

1h

3h

5h

crystallization criterionA. Secchi & M.R., PRB 82, 035417 (2010) A. Secchi & M.R., PRB 82, 035417 (2010)

a = WMb = particle-in-a-box

a

b

conclusions

Wigner molecules form in realistic samples

outlook (II)quantum devices (localization + spin-orbit coupling + electric control)

scanning tunneling spectroscopy

ww

w.n

an

oscie

nce

.un

imore

.it/max.h

tml

ww

w.n

an

o.c

nr.

it

nanotube quantum dots strongly correlated

graphene quantum dots

few-body physics of cold Fermi atomsM. Rontani et al., PRL 102, 060401 (2009)

Wigner molecules in carbon-nanotube quantum dots

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