1' dia

1
Non-equilibrium transport through quantum dots
B. Horváth, B. Lazarovits, O. Sauret, G. Zaránd
Department of Theoretical Physics, Budapest
University of Technology, H-1111,
Budapest, Hungary

• Abstract Systems like molecules between point
  contacts are often treated with the help of
  density functional theory, which treats the
  correlations essentially at the mean field level.
  We solved the nonequilibrium Anderson-model at
  mean field level the results show many
  unphysical properties such as hysteresis and
  spontaneous symmetry breaking.
• To study the artefacts of mean field
  approximation, we applied
• 1-level Anderson mean-field approximation
  (scattering states, Keldysh)
• 1-level Anderson 2nd order iterative
  perturbation theory 1
• 2-level Anderson mean-field approximation

2
Non-equilibrium Anderson model scattering states
Hamiltonian
Scattering states
Results from scattering states method
Self-consistent equations
Current
Renormalized energy level
3
Non-equilibrium Anderson model Keldysh
Keldysh Greens functions 2,3
Perturbation theory in
Unperturbed Greens functions
self-consistency
non-equilibrium
Keldysh Dysons equation
The mean field results with scattering states and
Keldysh-formalism are equivalent.
4
Non-equilibrium Anderson model MF-level results
non-physical results
? hysteresis
? sponteneous symmetry breaking
Exact result
5
Non-equilibrium Anderson model MF-level phase
diagrams
increasing bias decreasing bias
In G-µ plane
In ed-µ plane
non-physical results
? out of equilibrium magnetization
For higher T (TgtTc) magnetic phase disappears
? hysteresis
6
Non-equilibrium Anderson model
2nd order perturbation theory
Iterative perturbation theory 1 use MF Greens
functions as unperturbed Greens functions,
Hartree-type diagrams cancels.
First correction
Keldysh self-energies
calculating inserting to
FFT
7
Non-equilibrium Anderson model 2nd order
perturbation theory, symmetric case (ed-U/2)
Non-equilibrium spectral function in symmetric
case
In symmetric case 1/2
ed-1, U2, G0.2, T0
Effect of the bias voltage cancel the Kondo
peak, Hubbard peaks survive
8
Non-equilibrium Anderson model 2nd order
perturbation theory, asymmetric case (ed?-U/2)
Nonequilibrium spectral functions in th
asymmetric case (can be compared to4)
?1/2
The effect of the temparature is qualitatively
the same as the bias voltage
9
Two-level Anderson model
Singlet-triplet transition the ground state of
the two-level system can change from triplet to
singlet, with increasing separation between the
two level (?ed). If ?edltJ (exchange coupling),
the triplet state is favoured according to Hunds
rules.
Hamiltonian
where
U onsite Coulomb integral J Hunds rule
coupling
Vertex function
G
New terms, because of the number of levels
Components
10
Two-level Anderson model MF-level I.
First order perturbation (renormalize energy
levels)
G
Hartree-Fock Greens functions
Assumption
Occupation number of the different orbitals
Self-energy diagonal
No spin-polarization
11
Two-level Anderson model MF-level II.
In case of spin polarization, the self-energy
Occupation number for different spins and orbitals
Small ?ed up electrons in both levels Large ?ed
up-down electrons in lower level (upper is empty)
12
Conclusions and outlook

• Conclusions
• mean field artefacts disappeared within the
  second order iterative perturbation theory
• non-equilibrium spectral functions with proper
  equilibrium limit 4 bias voltage has the same
  effect as temperature
• two level Anderson-model is treated in mean
  field level sign of singlet-triplet transition
  can be seen
• Outlook
• apply iterative perturbation theory to the two
  level system
• apply fluctuation exchange approximation 5
• References
• 1 A. L. Yeyati et al., Phys. Rev. Lett. 71,
  2991
• 2 J. Rammer and H. Smith, Rev. Mod. Phys. 58,
  323
• 3 C. Caroli et al., J. Phys. C Solid State
  Phys. 5, 21
• 4 B. Horvatic et al., Phys. Rev. B 36, 675
• 5 N. E. Bickers and D. J. Scalapino, Ann. Phys.
  (N. Y.) 193, 206