1. Conductance of a spin-1 QD: two-stage Kondo effect Anna Posazhennikova Institut für Theoretische Festkörperphysik, Uni Karlsruhe, Germany Les Houches, June 19, 2006 Collaborators: Babak Bayani (TFP, University Karlsruhe, Germany) Piers Coleman (Rutgers University, NJ, USA)

2. Outline Introduction: Kondo effect in bulk and mesoscopic systems Spin-1 QD: theoretical expectations Model and T-matrix analysis Conductance calculations Conclusions and outlook

3. Introduction: Kondo Effect

4. History of Kondo effect System: metallic host + magnetic impurity 1930 – ρ min in some alloys 1950 – Χ curie – measurements showed that a LM forms in those alloys, which exhibit ρ min Q1: why does LM form? 1961 - P. W. Anderson: U is large enough compared to interlevel spacing Atomic limit of Anderson model Possible states

5. History of Kondo effect Q2: why does the formation of LM lead to ρ min ? 1964 – Jun Kondo, Hamiltonian Perturbation theory breaks down at T K T K is the only scale in the problem Q3: why does ρ saturates at low temperatures? 1970 – conjecture of Anderson and Yuval – GS is a paramagnetic spin singlet confirmed by 1971 - K.Wilson – NRG Up to here: orbital momentum

6. History of Kondo effect Q4: what happens in more realistic situation with 1980 Blandin, Nozieres 1984 Andrei, Tsvelik, Wiegman Perfectly Screened KEUnderscreened KEOverscreened KE FL NFL USK, OSK – inaccessible in bulk materials Mesoscopics?

7. Introduction: Kondo Effect in Quantum Dots

8. Introduction: Kondo Effect in QD

9. Realization of spin-1 QD: singlet-triplet transition in zero magnetic field N even N odd

10. Spin-1 QD: two-channel Kondo effect Set up: spin, coupled to left (L) and right (R) leads LR diagonalization J 1, J 2 – two coupling constants => two screening channels Kondo Hamiltonian

11. Conductance of a spin-1 QD: one-channel => two channel crossover Pustilnik, Glazman, PRL’01

12. Reminder: Kondo ModelAnderson Model Hubbard-Stratonovich transformation Schrieffer-Wolff transformation 2 channel Anderson H-an (infinite U)

13. Auxiliary particle representation of the Anderson Hamiltonian Introduce auxiliary particles + constraint Schwinger bosons auxiliary fermions (holons)

14. Novel large-N approximation

15. T-matrix for one channel and at finite voltage for different temperatures

16. Conductance of a spin-1 QD: expectations (reminder) One-channelTwo-channel Log correction due to USK!

17. Current through the dot Follow method proposed by Meir, Wingreen, PRL 1992 -Keldysh Green‘s functions

18. Current throught the dot Single-channel contributions + interference term Goal: calculation of the dot‘s Green‘s functions D

19. One-channel current Note: the dot GF is proportional to the t-matrix of conduction electrons

20. Results: conductance – one channel Linear conductance Voltage-dependence of conductance

21. Results: conductance two-channel QD Everything is messed up by the interchannel GFs => the current is not expressed In terms of t-matrices of single channels Simplifications: linear conductance – elastic scattering => results 1 2 3 T K2 /T K1 1 – 10 2 – 100 3 – 1000

22. Comparison with NRG Hofstetter, Schoeller, PRL 2002 NRG gives qualitatively same results

23. Conclusions and future work We calculated analytically transport in a spin-1 QD in case of one and two-channels In case of one –channel singular conductance is obtained – signature of US Kondo effect In case of two-channels interference effects are observed – conductance is suppressed at low T Future projects Improve the results for two-channel conductance, taking into account the inelastic scattering terms Inclusion of magnetic field Inclusion of spin – relaxation effects