Coupled quantum dots:a laboratory for studying quantum impurity physics

Rok Žitko

SISSA, Trieste, 30. 10. 2007

Jožef Stefan Institute, Ljubljana, Slovenia

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 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:

http://nrgljubljana.ijs.si/

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, 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

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).

triple-dot device

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

-2 -1 A 1 2

B

Two-stage Kondo effect

R. Žitko, J. Bonča: Enhanced conductance through side-coupled double quantum dots, Phys. Rev. B 73, 035332 (2006).

-2 -1 A 1 2

B

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).

A. Ramšak, J. Mravlje, R. Žitko, J. Bonča:Spin qubits in double quantum dots - entanglement versus the Kondo effectPhys. Rev. B 74, 241305(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, 045312 (2006)

Vk = eikL vkVk≡V (L0)

Effective single impurity S=N/2 Kondo model

The RKKY interaction is ferromagnetic, JRKKY>0:

S is the collective S=N/2 spin operator of the coupled impurities,

S=P(Si)P

Effective model (T<JRKKY):

JRKKY0.62 U(0JK)2 4th order perturbation in Vk

Free orbital regime

(FO)

Local moment regime

(LM)

Ferro-magnetically frozen (FF)

Strong-coupling

regime (SC)

o o

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 transition1=+, 2=-

S=1 KondoS=1/2 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)

J t

Good agreement between 3 methods:

• CPMC – constrained path quantum Monte CarloZhang, Carlson and Gubernatis, PRL 74, 3652 (1995); PRB Zhang, Carlson and Gubernatis, PRL 74, 3652 (1995); PRB 5959, 12788 (1999)., 12788 (1999).

• GS – projection/variational method.

Schonhammer, Z. Phys. B Schonhammer, Z. Phys. B 2121, 389 (1975); PRB , 389 (1975); PRB 1313, 4336 (1976), Gunnarson and , 4336 (1976), Gunnarson and Schonhammer, PRB Schonhammer, PRB 3131, 4185 (1985), Rejec and Ram, 4185 (1985), Rejec and Ramššak, PRB 68, 035342 (2003).ak, PRB 68, 035342 (2003).

• NRG – numerical renormalization group

Krishna-murthy, Wilkins and Wilson, PRB Krishna-murthy, Wilkins and Wilson, PRB 2121, 1003 (1980); Costi, Hewson and Zlati, 1003 (1980); Costi, Hewson and Zlatićć, J. , J. Phys.: Condens. Matter Phys.: Condens. Matter 66, 2519, (1994)., 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

non-Fermi liquid

• Gserial=G0

Fermi liquid

See also: G. Zaránd et al. PRL 97, 166802 (2006).

TK(1)

TK(2)

T

NFL

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.

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., 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