Coupled quantum dots: a laboratory for studying quantum impurity physics Rok Žitko SISSA, Trieste, Jožef Stefan Institute, Ljubljana, Slovenia

Co-workers Quantum transport theory –prof. Janez Bonča 1,2 –prof. Anton Ramšak 1,2 –Tomaž Rejec 1,2 –Jernej Mravlje 1 Experimental surface science and STM –prof. Albert Prodan 1 –prof. Igor Muševič 1,2 –Erik Zupanič 1 –Herman van Midden 1 –Ivan Kvasić 1 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 single- atom 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:

W. G. van der Wiel, S. de Franceschi, T. Fujisawa, J. M. Elzerman, S. Tarucha, L. P. Kouwenhoven, Science 289, 2105 (2000) Conduction as a function of gate voltage for decreasing temperature Kondo effect in quantum dots

Scattering theory “Landauer formula” See, for example, M. Pustilnik, L. I. Glazman, PRL 87, (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 L. Gaudreau, S. A. Studenikin, A. S. Sachrajda, P. Zawadzki, A. Kam, J. Lapointe, M. Korkusinski, and P. Hawrylak, Phys. Rev. Lett. 97, (2006). M. Korkusinski, I. P. Gimenez, P. Hawrylak, L. Gaudreau, S. A. Studenikin, A. S. Sachrajda, Phys. Rev. B 75, (2007). triple-dot device

Systems of coupled quantum dots and “exotic” types of the Kondo effect

Two-stage Kondo effect R. Žitko, J. Bonča: Enhanced conductance through side-coupled double quantum dots, Phys. Rev. B 73, (2006). See also: P. S. Cornaglia, D. R. Grempel, PRB 71, (2005) M. Vojta, R. Bulla, W. Hofstetter, PRB 65, (R) (2002).

For J<T K, Kondo screening occurs in two steps. T K (1) T K (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, (2006).

A. Ramšak, J. Mravlje, R. Žitko, J. Bonča: Spin qubits in double quantum dots - entanglement versus the Kondo effect Phys. Rev. B 74, (R) (2006) The inter-impurity spin entanglement vs. the Kondo effect

Parallel quantum dots and the N-impurity Anderson model R. Žitko, J. Bonča: Multi-impurity Anderson model for quantum dots coupled in parallel, Phys. Rev. B 74, (2006) V k = e ikL v k V k ≡V (L  0)

Effective single impurity S=N/2 Kondo model The RKKY interaction is ferromagnetic, J RKKY >0: S is the collective S=N/2 spin operator of the coupled impurities, S=P(  S i )P Effective model (T<J RKKY ): J RKKY  0.62 U(  0 J K ) 2 4 th order perturbation in V k

Free orbital regime (FO) Local moment regime (LM) Ferro- magnetically frozen (FF) Strong- coupling regime (SC)

The spin-N/2 Kondo effect Full line: NRGSymbols: Bethe Ansatz

Discontinuities in G  quantum phase transitions

Chrage fluctuations vs. ferromagnetic alignment first-order transition

Kondo modelKondo 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, (2005).

Kosterlitz-Thouless transition  1 =+ ,  2 =-  S=1 Kondo S=1/2 Kondo

Triple quantum dot R. Žitko, J. Bonča, A. Ramšak, T. Rejec: Kondo effect in triple quantum dot, Phys. Rev. B 73, (2006) R. Žitko, J. Bonča: Fermi-liquid versus non-Fermi-liquid behavior in triple quantum dots, Phys. Rev. Lett. 98, (2007)

J  t Good agreement between 3 methods: CPMC – constrained path quantum Monte Carlo Zhang, Carlson and Gubernatis, PRL 74, 3652 (1995); PRB 59, (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, (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).

G side ~G 0 /2, G serial ~0  non-Fermi liquid G serial =G 0  Fermi liquid See also: G. Zaránd et al. PRL 97, (2006). T K (1) T K (2) TT NFL

CFT prediction: 0, 1/8, 1/2, 5/8, 1, 1+1/8,...

Conductance: quantum dots in series N=2 N=3N=4 See also: A. Oguri, Y. Nisikawa and A. C. Hewson, J. Phys. Soc. Japan, (2005). Y. Nisikawa, A. Oguri. Phys. Rev. B 73, (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 Cu/Cu(111) at T=10 K.

Scanning tunneling spectroscopy: we measure local density of states, i.e. spectral functions. STM tip metal surface Fano resonance in STS spectra due to Kondo effect in Co ions on various surfaces. [P. Wahl et al., Phys. Rev. Lett., , 2004]

Two-impurity Kondo problem on surfaces P. Wahl et al., Phys. Rev. Lett. 98, (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