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Coulomb versus spin-orbit interaction
in carbon-nanotube quantum dots
Andrea Secchi and Massimo Rontani
CNR-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
F. Kuemmeth et al., Nature 452, 448 (2008)
double degeneracy
Strong correlation or not in CN QDs?
Strong correlation or not in CN QDs?
Low temperature SETS experiment
spin-orbit interaction
splits 4-fold degenerate
spin-orbitals
spin
isospin
Strong correlation or not in CN QDs?
two-electron ground state:
one Slater determinant
no correlation
chemical potential
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: e
M. Rontani et al., JCP 124, 124102 (2006)
Strongly correlated CI wave functions
different harmonic oscillator quantum numbers
A & B states:
strongly correlated
same orbital wave
functions
differ in isospin only
isospin =
valley population
A. Secchi and M. Rontani, arXiv: 0903.5107
Independent-particle feature explained
exp
theo
A and B:
N=2
N=1
T3 = 0
T3 = 1
correlated
T3 = 0, 1
T3 = -1/2
T3 = 1/2
split by spinorbit int. only
B(T)
A. Secchi and M. Rontani, arXiv: 0903.5107
Non-universal tunneling spectrum
exp
0  8meV
0  4meV
N=2
N=1
A. Secchi and M. Rontani, arXiv: 0903.5107
CI two-electron energy spectrum
ungerade
n(x)
0
x
gerade
A. Secchi and M. Rontani, arXiv: 0903.5107
Pair correlation functions
g(X) = probability to find a couple of electrons at relative distance X

dCA 
 g X  g X  dX
C
0
A
Conclusions
• spin-orbit and Coulomb interactions coexist
• non-interacting features of tunneling spectra explained
• we predict electrons to form a Wigner molecule
[email protected]
[email protected]
www.s3.infm.it
www.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
Hˆ SO  ( SO / R)ˆ zˆz
Wigner-Mattis theorem is not appliable in nanotubes
( x1 , z1; x2 ,  z 2 )   ( x1 , x2 )  ( z1 , z 2 )
 ( x1 , x2 )
nodeless in the ground state
 ( x1, x2 )   ( x2 , x1 )
 ( z1, z 2 )   ( z 2 , z1 )
S=0
isospin T = additional degree of freedom
( x1 , z1 ,  z1; x2 , z 2 ,  z 2 )   ( x1 , x2 ) ( z1 , z 2 )  ( z1 , z 2 )
either (S = 0, T = 1) or (S = 1, T = 0)
Tz = -1, 0, +1
Sz = -1, 0, +1