Kondo effect in a quantum dot without spin

1. Kondo effect in a quantum dot without spin Hyun-Woo Lee (Postech) & Sejoong Kim (Postech  MIT) References: H.-W. Lee & S. Kim, cond-mat/0610496 P. Silvestrov & Y. Imry, cond-mat/0609355 V. Kashcheyevs, A. Schiller, A. Aharony, & O. Entin-Wohlman, cond-mat/0610194

2. Kondo effect • Temperature dependence of resistance • Resistance minimum

3. Before After After Kondo effect (continued)[J. Kondo, Prog. Theor. Phys. ’64] • Scattering by magnetic impurities s-d model

4. LogT dependence in R(T) (*) 엄종화 Scattering amplitude for a channel of where N : number of d-electrons, N(0) : density of state at EF D : width of conduction electron distribution around EF Jk,q = J where –D < ek, eq < D = 0 otherwise This lnT dependence combined with the phonon contribution (T5 dependence) makes a resistance minimum in R(T).

5. Kondo effect (continued) • High T vs. low T Kondo singlet Cf. Asymptotic freedom

6. TK << T일 때, (*) 엄종화 Kondo effect When T << TK, r ~ r0 - cT2 : unitary limit TK ~ T일 때, Hamann expression (Phys. Rev. 1967) For TK > T, take (-) in the equation TK < T, take (+) in the equation

7. Kondo effect in AuFe(26ppm) wire (*) 엄종화 Hamann expression (Phys. Rev. 1967) From fitting the Hamann expression to r(T), we obtain S = 0.12, TK = 0.99 K. Slope of Kondo resistivity = 0.11 nWcm / (ppm decade K) Concentration of AuFe is estimated by the slope of Dr => 26 ppmin the above figure

8. Kondo effect in quantum dot

9. n Vg Quantum dot (QD) • “Metallic” limit ~e2/2C >> kT

10. Transport through a QD • Orthodox theory of Coulomb blockade • Transport due to charge fluctuations

11. E>> kT Quantum confinement • Single particle energy quantization

12. n 0 3 1/2 2 0 1 S=1/2 Vg Even-odd effect • Spin singlet (S=0) vs doublet (S=1/2) • QD with odd n = magnetic impurity ???

13. c.f. n After Before Vg Kondo effect in QD ? • Hamiltonian • Spin flip via second order processes

14. Kondo effect in QD w/ odd n  Kondo suppression ofR • Theories • T. K. Ng and P. A. Lee • Phys. Rev. Lett. 61, 1768 (1988) • L. I. Glazman and M. E. Raikh • JETP Lett. 47, 452 (1988) • Experiments • D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abush-Magder, U. Meirav, and M. A. Kaster • Nature 391, 156 (1998); Phys. Rev. Lett. 81, 5225 (1998) • S. M. Cronenwett, T. H. Oostercamp, and L. P. Kouwenhoven • Science 281, 540 (1998)

15. Unitary limit of the Kondo effect in SET [W. G. van der Wiel et al., Science ’00] (*) 엄종화 온도대역: 15 mK – 800 mK G(T) G at Vgl = -413mV shows logarithmic T dependence (inset), and saturates below 90mK (unitary limit) This experiment shows a unitary limit = 2e2/h (GR=GL의 경우) Kondo resonance peak Vgl was fixed at -413mV. VSD was biased between S and D. FWHM

16. Kondo temperature: TK (*) 엄종화 In Anderson model, ; Costi et al., J. Phys.: Condense. Matter 6, 2519 (1994) TK in Log scale An empirical function ; Goldhaber-Gordon et al., PRL 81, 5225 (1998) : universal functional form of T/TK s is a fit parameter, but is almost constant ~0.2 in the Kondo regime.

17. Kondo effect in QD w/o spin?

18. t1L t1R t t t1R t2L Two level QD • QD w/ two single-particle level • Source & Drain • Tunneling • “Spin” ?

19. Pseudospin for 1=2(=) • Unitary transformations Pseudospin up Pseudospin Pseudospin down

20. 0 1 2 Schrieffer-Wolf transformation:QD system (Anderson model)  s-d model • Fock space decomposition • Full Hamiltonian • Projection to n=1 Fock space

21. Effective Hamiltonian Hs-d • Total Hamiltonian • Anisotropic antiferro-exchange • U(1) instead of SU(2) • Pseudomagnetic field Bzeff • (*) For = • SU(2): Jz=J+=J- • Bzeff=0

22. hz 0 1 2 =+U/2 0 -U/2 Pseudomagnetic field Bzeff • Expectation value • For   >  •  Population switching from level to  level with decreasing 

23. 0 1 2 =+U/2 0 -U/2 Charge 10 Charge 12 U U Population switching (PS) [Silvestrov & Imry, PRL’00] • Energy renormalization • eff= bare+  (hopping) •  : gate voltage dependent

24. D D Poor man’s scaling • [1] Fock space decomposition • [2] Full Hamiltonian • [3] Projection to “g” sector of Fock space • New Hamiltonian w/ reduced D • [4] Back to [1] D

25. Scaling equations • Exchange J’s • Scaling invariant • Integration: Characteristic energy scale (Kondo temperature) • Pseudomagnetic field Bzeff • Integration:

26. Anisotropic s-d model • Approximation •  Anisotropic s-d model • Exact solution (via Bethe ansatz) available !!! • Tsvelick & Wiegmann, Adv. Phys. 32, 453 (1983)

27. Conductance at T=0 •  and  scattering states • Friedel sum rule • Landauer-Buttiker formula

28. Anisotropic s-d model [Tsvelick and Wiegmann, Adv. Phys. (1983)] • Sz=(n-n)/2 vs. hz • G vs. 

29. 0 1 2 =+U/2 0 -U/2 Cf. Conventional spin Kondo • Conventional spin Kondo • Kondo w/o spin • Correlation-induced resonance [Meden & Marquardt, PRL 96, 146801 (2006)]

30. w/o degeneracy 1-2 0 • Same Unitary transformation • Additional pseudomagnetic field h • Parallel to z • Shift of CIRs • Perpendicular to z • Asymmetry in CIRs (Fano-like)

31. Summary • Kondo effect in QD w/o spin • Distinct conductance pattern (cf. spin Kondo in QD) • Future directions • w/o degeneracy • Temperature dependence • Pseudospin & real spin • Real Spin [SU(2)] • Pseudospin [Not SU(2) invariant] • Connection w/ anomalous transmission phase problem ?