Coupled quantum dots: a laboratory for studying quantum impurity physics

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Coupled quantum dotsa laboratory for studying
quantum impurity physics

• Rok Žitko

Jožef Stefan Institute, Ljubljana, Slovenia
SISSA, Trieste, 30. 10. 2007
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Co-workers

• Experimental surface science and STM
• prof. Albert Prodan1
• prof. Igor Muševic1,2
• Erik Zupanic1
• Herman van Midden1
• Ivan Kvasic1

• Quantum transport theory
• prof. Janez Bonca1,2
• prof. Anton Ramšak1,2
• Tomaž Rejec1,2
• Jernej Mravlje1

1 Jožef Stefan Institute, Ljubljana, Slovenia 2
Faculty of Mathematics and Physics, University of
Ljubljana, Ljubljana, Slovenia
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Transport in nanostructures
Cu/Cu(111)IJS, 2007
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Outline

• Kondo physics in quantum dots
• Coupled quantum dots as impurity clusters
• side-coupled double QD and two-stage Kondo
  effect
• N parallel QDs (N1...5, one channel) and quantum
  phase transitions
• N serial QDs (N14, two channels) and non-Fermi
  liquid physics
• Low-temperature STM manipulations and
  single-atom spectroscopy

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Tools SNEG and NRG Ljubljana
Add-on package for the computer algebra system
Mathematica for performing calculations involving
non-commuting operators

• Efficient general purpose numerical
  renormalization group code
• flexible and adaptable
• highly optimized (partially parallelized)
• easy to use

Both are freely available under the GPL
licence http//nrgljubljana.ijs.si/
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Kondo effect in quantum dots
Conduction as a function of gate voltage for
decreasing temperature
W. G. van der Wiel, S. de Franceschi, T.
Fujisawa, J. M. Elzerman, S. Tarucha, L. P.
Kouwenhoven, Science 289, 2105 (2000)
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Scattering theory
Landauer formula
See, for example, M. Pustilnik, L. I. Glazman,
PRL 87, 216601 (2001).
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Keldysh approach
One impurity
Y. Meir, N. S. Wingreen. PRL 68, 2512 (1992).
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Conductance of a quantum dot (SIAM)
Computed using NRG.
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Systems of coupled quantum dots
triple-dot device
L. Gaudreau, S. A. Studenikin, A. S. Sachrajda,
P. Zawadzki, A. Kam, J. Lapointe, M. Korkusinski,
and P. Hawrylak,Phys. Rev. Lett. 97, 036807
(2006). M. Korkusinski, I. P. Gimenez, P.
Hawrylak,L. Gaudreau, S. A. Studenikin, A. S.
Sachrajda,Phys. Rev. B 75, 115301 (2007).
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Systems of coupled quantum dots and exotic
types of the Kondo effect
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Two-stage Kondo effect
R. Žitko, J. Bonca Enhanced conductance through
side-coupled double quantum dots, Phys. Rev. B
73, 035332 (2006).
See also P. S. Cornaglia, D. R. Grempel, PRB 71,
075305 (2005)M. Vojta, R. Bulla, W. Hofstetter,
PRB 65, 140405(R) (2002).
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For JltTK, Kondo screening occurs in two steps.
TK(1)
TK(2)
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Spin-charge separation ? Simultaneous spin and
charge Kondo effects
R. Žitko, J. Bonca Spin-charge separation and
simultaneous spin and charge Kondo effect, Phys.
Rev. B 74, 224411 (2006).
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The inter-impurity spin entanglement vs. the
Kondo effect
A. Ramšak, J. Mravlje, R. Žitko, J. Bonca Spin
qubits in double quantum dots - entanglement
versus the Kondo effect Phys. Rev. B 74,
241305(R) (2006)
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Parallel quantum dots and the N-impurity
Anderson model
Vk eikL vk
VkV (L?0)
R. Žitko, J. Bonca Multi-impurity Anderson model
for quantum dots coupled in parallel, Phys. Rev.
B 74, 045312 (2006)
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Effective single impurity SN/2 Kondo model
The RKKY interaction is ferromagnetic, JRKKYgt0
JRKKY??0.62 U(r0JK)2
4th order perturbation in Vk
Effective model (TltJRKKY)
S is the collective SN/2 spin operator of the
coupled impurities, SP(SSi)P
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Free orbital regime (FO)
Ferro-magnetically frozen (FF)
Local moment regime (LM)
Strong-coupling regime (SC)
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The spin-N/2 Kondo effect
Full line NRG
Symbols Bethe Ansatz
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Discontinuities in G ? quantum phase transitions
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Chrage fluctuations vs. ferromagnetic alignment
first-order transition
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Kondo model
Kondo model potential scattering
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S1 Kondo model potential scattering
S1/2 Kondo model strong potential scattering
S1 Kondo model
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Gate-voltage controlled spin filtering
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Local occupancy variation
Occupancy switching G-dependent coupling vs.
charging energy U
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Spectral functions - underscreening
See also A. Posazhennikova, P. Coleman, PRL 94,
036802 (2005).
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Kosterlitz-Thouless transition
d1D, d2-D
S1/2 Kondo
S1 Kondo
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Triple quantum dot
R. Žitko, J. Bonca, A. Ramšak, T. Rejec Kondo
effect in triple quantum dot, Phys. Rev. B 73,
153307 (2006) R. Žitko, J. Bonca Fermi-liquid
versus non-Fermi-liquid behavior in triple
quantum dots, Phys. Rev. Lett. 98, 047203 (2007)
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• Good agreement between 3 methods
• CPMC constrained path quantum Monte Carlo
• Zhang, Carlson and Gubernatis, PRL 74, 3652
  (1995) PRB 59, 12788 (1999).
• GS projection/variational method.
• Schonhammer, Z. Phys. B 21, 389 (1975) PRB 13,
  4336 (1976), Gunnarson and Schonhammer, PRB 31,
  4185 (1985), Rejec and Ramšak, PRB 68, 035342
  (2003).
• NRG numerical renormalization group
• Krishna-murthy, Wilkins and Wilson, PRB 21, 1003
  (1980) Costi, Hewson and Zlatic, J. Phys.
  Condens. Matter 6, 2519, (1994).

J ? t
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Non-Fermi liquid behavior of the two-channel
Kondo model type
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Two-channel Kondo model
Experimental observation R. M. Potok et al.,
Nature 446, 167 (2007).
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• GsideG0/2, Gserial0
• ? non-Fermi liquid
• GserialG0
• ? Fermi liquid

TK(1)
See also G. Zaránd et al. PRL 97, 166802 (2006).
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CFT prediction 0, 1/8, 1/2, 5/8, 1, 11/8, ...
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Conductance quantum dots in series
N2
N3
N4
See also A. Oguri, Y. Nisikawa and A. C. Hewson,
J. Phys. Soc. Japan, 74 2554 (2005).Y. Nisikawa,
A. Oguri. Phys. Rev. B 73, 125108 (2006).
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Low-temperature STM (2004)
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Besocke beetle Working temperature 5.9 K
Gerhard Meyer (FU Berlin, now at IBM Research
Division, Rüschlikon)Stefan Fölsch (Paul Drude
Institute, Berlin)SPS-Createc GmbH
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High mechanical stability!
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Erik Zupanic, IJS, July 2007. Cu/Cu(111) at T10
K.
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STM tip
metal surface
Scanning tunneling spectroscopy we measure
local density of states, i.e. spectral functions.
Fano resonance in STS spectra due to Kondo effect
in Co ions on various surfaces.
P. Wahl et al., Phys. Rev. Lett., 93 176603,
2004
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Two-impurity Kondo problem on surfaces
P. Wahl et al., Phys. Rev. Lett. 98, 056601
(2007).
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Conclusions and outlook

• Impurity clusters can be systematically studied
  with ease using flexible NRG codes
• Very rich physics various Kondo regimes, quantum
  phase transitions, etc. But to what extent can
  these effects be experimentally observed?
• Towards more realistic models better description
  of inter-dot interactions, role of QD shape and
  distances.
• Surface Kondo effect in clusters of two or three
  magnetic adatoms
• low-temperature high-field experimental studies
• DFT NRG study