coulomb versus spin-orbit interaction in carbon-nanotube quantum dots andrea secchi and massimo...
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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?
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
Conclusions
• spin-orbit and Coulomb interactions coexist
• non-interacting features of tunneling spectra explained • we predict electrons to form a Wigner molecule
[email protected]@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
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