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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.

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## Coulomb versus spin-orbit interaction in carbon-nanotube quantum dots

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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?**Low temperature SETS experiment spin-orbit interaction splits 4-fold degenerate spin-orbitals spin isospin**Strong correlation or not in CN QDs?**the simplest interpretation two-electron ground state: one Slater determinant no correlation chemical potential**CI model: 1D harmonic potential**configuration-interaction (CI) calculation: two valleys QD: harmonic potential forward & backward Coulomb interactions spin-orbit coupling free parameter: e theory exp 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**A and B:**correlated T3 = 0, 1 split by spin-orbit int. only T3 = 1 T3 = 0 T3 = -1/2 T3 = 1/2 Independent-particle feature explained exp theo N = 2 N = 1 B(T) A. Secchi and M. Rontani, arXiv: 0903.5107**Non-universal tunneling spectrum**exp N = 2 N = 1 A. Secchi and M. Rontani, arXiv: 0903.5107**n(x)**x CI two-electron energy spectrum ungerade 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**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.it massimo.rontani@unimore.it 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**six-fold degenerate**Spin-orbit coupling for two electrons**isospin T = additional degree of freedom**either (S = 0, T = 1) or (S = 1, T = 0) Tz = -1, 0, +1 Sz = -1, 0, +1 Wigner-Mattis theorem is not appliable in nanotubes nodeless in the ground state S = 0

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