Transcript Document

Coupled quantum dots:
a laboratory for studying
quantum impurity physics
Rok Žitko
Jožef Stefan Institute,
Ljubljana, Slovenia
SISSA, Trieste, 30. 10. 2007
Co-workers
• Quantum transport theory
– prof. Janez Bonča1,2
– prof. Anton Ramšak1,2
– Tomaž Rejec1,2
– Jernej Mravlje1
• Experimental surface science and
STM
– prof. Albert Prodan1
– prof. Igor Muševič1,2
– Erik Zupanič1
– Herman van Midden1
– Ivan Kvasić1
1
Jožef Stefan Institute, Ljubljana, Slovenia
2
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
Transport in nanostructures
Cu/Cu(111)
IJS, 2007
Outline
• Kondo physics in quantum dots
• Coupled quantum dots as impurity clusters:
– side-coupled double QD and
two-stage Kondo effect
– N parallel QDs (N=1...5, one channel) and quantum
phase transitions
– N serial QDs (N=1…4, two channels) and
non-Fermi liquid physics
• Low-temperature STM: manipulations and singleatom spectroscopy
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/
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)
Scattering theory
“Landauer formula”
See, for example, M. Pustilnik, L. I. Glazman, PRL 87, 216601 (2001).
Keldysh approach
One impurity:
Y. Meir, N. S. Wingreen. PRL 68, 2512 (1992).
Conductance of a quantum dot (SIAM)
Computed using NRG.
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).
Systems of coupled quantum dots and
“exotic” types of the Kondo effect
-2
-1
A
B
1
2
Two-stage Kondo effect
-2
-1
A
1
2
B
R. Žitko, J. Bonča: 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).
For J<TK, Kondo
screening occurs
in two steps.
TK(1)
TK(2)
Spin-charge separation  Simultaneous
spin and charge Kondo effects
R. Žitko, J. Bonča: Spin-charge separation and simultaneous spin and charge Kondo
effect, Phys. Rev. B 74, 224411 (2006).
The inter-impurity spin entanglement vs. the Kondo effect
A. Ramšak, J. Mravlje, R. Žitko, J. Bonča:
Spin qubits in double quantum dots - entanglement versus the Kondo effect
Phys. Rev. B 74, 241305(R) (2006)
Parallel quantum dots and
the N-impurity Anderson model
ikL v
=
e
VVk≡V
(L0)
k
k
R. Žitko, J. Bonča: Multi-impurity Anderson model for quantum dots coupled in parallel,
Phys. Rev. B 74, 045312 (2006)
Effective single impurity S=N/2
Kondo model
The RKKY interaction is ferromagnetic, JRKKY>0:
JRKKY0.62 U(r0JK)2
4th order perturbation in Vk
Effective model (T<JRKKY):
S is the collective
S=N/2 spin operator of
the coupled impurities,
S=P(SSi)P
Free orbital
regime
(FO)
Local
moment
regime
(LM)
o
o
Ferromagnetically
frozen (FF)
Strongcoupling
regime (SC)
The spin-N/2 Kondo effect
Full line: NRG
Symbols: Bethe Ansatz
Discontinuities in G  quantum phase transitions
Chrage fluctuations vs.
ferromagnetic alignment
first-order transition
Kondo model
Kondo model +
potential scattering
S=1 Kondo
model
S=1 Kondo
model +
potential
scattering
S=1/2 Kondo
model +
strong potential
scattering
Gate-voltage controlled spin filtering
Local occupancy variation
Occupancy switching: Γ-dependent coupling vs. charging energy U
Spectral functions - underscreening
See also: A. Posazhennikova, P. Coleman, PRL 94, 036802 (2005).
Kosterlitz-Thouless transition
d1=+D, d2=-D
S=1/2 Kondo
S=1 Kondo
Triple quantum dot
R. Žitko, J. Bonča, A. Ramšak, T. Rejec: Kondo effect in triple quantum dot,
Phys. Rev. B 73, 153307 (2006)
R. Žitko, J. Bonča: Fermi-liquid versus non-Fermi-liquid behavior in triple quantum
dots, Phys. Rev. Lett. 98, 047203 (2007)
Good agreement between 3 methods:
• CPMC – constrained path
Monte Carlo
J quantum
t
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 Zlatić, J.
Phys.: Condens. Matter 6, 2519, (1994).
Non-Fermi liquid
behavior of
the two-channel
Kondo model type
Two-channel Kondo model
Experimental observation: R. M. Potok et al., Nature 446, 167 (2007).
• Gside~G0/2, Gserial~0
TK(1)
 non-Fermi liquid
• Gserial=G0
TK(2)
 Fermi liquid
TD
NFL
See also: G. Zaránd et al. PRL
97, 166802 (2006).
CFT prediction: 0, 1/8, 1/2, 5/8, 1, 1+1/8, ...
Conductance: quantum dots in series
N=2
N=3
N=4
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).
Low-temperature
STM
(2004)
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
High mechanical
stability!
Erik Zupanič, IJS, July 2007. Cu/Cu(111) at T=10 K.
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]
Two-impurity
Kondo problem on
surfaces
P. Wahl et al.,
Phys. Rev. Lett. 98,
056601 (2007).
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