wigner molecules in carbon-nanotube quantum dots massimo rontani and andrea secchi s3, istituto di...
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![Page 1: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/1.jpg)
Wigner moleculesin carbon-nanotube quantum dots
Massimo Rontani and Andrea SecchiS3, Istituto di Nanoscienze – CNR, Modena, Italy
![Page 2: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/2.jpg)
ultraclean semiconducting nanotubes
Bockrath group, Nature Phys. 2008 McEuen group, Nature 2008
gate-defined quantum dots
shallow confinement potentials (approx. parabolic)
![Page 3: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/3.jpg)
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
![Page 4: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/4.jpg)
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
![Page 5: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/5.jpg)
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()( NN spin added electron
![Page 6: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/6.jpg)
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()( NN isospin added el.(angular momentum)
![Page 7: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/7.jpg)
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
ground statespin & isospinpolarized
Wigner molecule?
single-particle + spin-orbit
![Page 8: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/8.jpg)
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)
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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
![Page 10: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/10.jpg)
experimental evidence
split 4-fold degenerate spin-orbitals
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non-interacting physics?
two-electron ground state:
one Slater determinant
no correlation chem
ical
pote
nti
al
the simplest interpretation
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theory vs experiment
theoryPRB 80, 041404(R) (2009) McEuen group 2008
B (T)
dielectric constant
fitting parameter
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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
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spectrum affected by interaction
N = 2
N = 1
A. Secchi & M.R., PRB 80, 041404(R) (2009)
interaction strength
SO
SO
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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
![Page 16: Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy](https://reader036.vdocuments.net/reader036/viewer/2022062322/5697bf711a28abf838c7df3d/html5/thumbnails/16.jpg)
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
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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)