Applications of atomic manybody methods:

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Applications of atomic many-body methods high-Z
ions and quantum dots Steven Blundell CEA-Grenobl
e, France
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Outline
QED in high-Z Zn-like ions
relativistic many-body perturbation theory
(with Walter, U. Safronova, and M. Safronova)
one- and two-body self-energy and vacuum
polarization effects
Atomic structure methods applied to semiconductor
quantum dots
basis sets for quasi-2D electronic systems
Wigner localization at low densities
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Zn-like and Cu-like ions
28-electron closed-shell Ni-like core
1s22s22p63s23p63d10 Cu-like single valence
electron Zn-like two valence electrons
Perturbative expansion around Dirac-Fock or
Breit-Dirac-Fock potential of the Ni-like
core Relativistic many-body perturbation
theory (RMBPT) for Cu-like ions W. R.
Johnson, S. A. Blundell, and J. Sapirstein, PRA
42, 1087 (1990) Coulomb correlation through
3rd order, Breit correlation through 2nd order
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Many-electron QED effects in Cu-like ions
S. A. Blundell, Phys. Rev. A 47, 1790 (1993)
28-electron Ni-like core screens self-energy
and vacuum polarization, Z ? Zeff Z c
? evaluate lowest-order SE and VP in a local
potential
V Vnuc VDF,dir
v valence c core
Valence-core exchange interaction
Core relaxation
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4s-4p3/2 transition in Cu-like ions
Theory S. A. Blundell, Phys. Rev. A 47, 1790
(1993) M. H. Chen, K. T. Cheng, W. R. Johnson,
and J. Sapirstein, Phys. Rev. A 74, 042510
(2006) Expt. E. Träbert, P. Beiersdorfer, and
H. Chen, Phys. Rev. A 70, 032506 (2004)
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Zn-like ions relativistic correlation

• Two valence electrons outside 28-electron
  closed-shell Ni-like core 1s22s22p63s23p63d10
• Valence space 4s, 4p1/2, 4p3/2, 4d3/2,
  4d5/2, 4f5/2, 4f7/2
• Treat as quasi-degenerate
• Hybrid RMBPT / CI formalism
• Heff C E C

Exact energies
Effective Hamiltonian
Heff 8 x 8 matrix (non-symmetric), 64
distinct matrix elements
RMBPT S. A. Blundell, W. R. Johnson, M.
Safronova, U. Safronova, PRA (2008)
Coulomb and Breit correlation through second order
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Many-electron QED effects in Zn-like ions
Treat self-energy and vacuum polarization terms
as effective interactions within RMBPT
Cu-like QED (SE, VP, valence-core exchange,
core relaxation) are one-body effects and
enter on diagonal of effective Hamiltonian
Valence-valence screening terms
Two-body effects, enter on diagonal and
off-diagonal parts of effective Hamiltonian
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4s2 1S0 4s4p 1P1 transition in Zn-like ions
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4s2 1S0 4s4p 1P1 transition in Zn-like ions
Expt. E. Träbert, P. Beiersdorfer, and H. Chen,
Phys. Rev. A 70, 032506 (2004)
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Semiconductor quantum dots
Quasi-2D electron system
system is always in lowest subband for
z-direction
effective 2D wavefunction
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Basic approximations
Electronic structure methods for 100 electrons
in well
Quasi-2D system
(Single-band) effective mass approximation for
electrons in conduction band
Electronic structure methods
Spin-density functional theory
Hartree-Fock, basis sets, configuration
interaction method, many-body theory
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Applications
Single-electron transistors, quantum point
contacts
Quantum transport
Intense magnetic fields quantum Hall regime
Regime of low electron density Wigner
localization
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Wigner molecules
Finite-size 2D analog of Wigner crystal in 3D
(1934)
No experimental realization (yet)
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Circular parabolic dots at low density N 6
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Pair function internal structure of wavefunction
spin-spatial pair correlation function
(conditional probability)
g(r0? r?) N 6, rs 12 a0 ground
state 1S
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Full CI for N 6 at low electronic density
low-lying states
1 A. Ghosal, A. D. Güçlü, C. J. Umrigar, D.
Ullmo, and H. U. Baranger, Phys. Rev. B (2007)
Nature (2006) Diffusion Monte Carlo
(fixed-node approximation) rs 18
a0, lowest-energy state with a given symmetry
(Lz, S) 2 M. Rontani, C. Cavazzoni, D. Bellucci,
and G. Goldoni, J. Chem. Phys. (2006)
CI with simple-harmonic-oscillator basis set 3
R. Egger, W. Hausler, C. H. Mak, and H. Grabert,
Phys. Rev. Lett. (1999)
Path-integral Monte-Carlo
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Excited states spin-spin, rotational,
vibrational, isomeric
g(r0? r?) N 6, rs 12 a0 states
with symmetry (Lz, S) (0, 3)
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Addition energy
Tarucha et al. (1996)
Theory (SDFT) Austing et al. (1999)
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Hunds rule single-particle shell splittings
Competition between exchange interaction and
single-particle level splittings